L'Hôpital's Rule is used to evaluate a limit when other techniques will not work, like factoring and rationalizing. It especially convenient to use with exponentials and, sometimes, limits involving trig functions. In fact, L'Hôpital's Rule can usually be used on limits where those other techniques work and it is often easier than factoring. So it is a nice tool to have when evaluating limits.  
L'Hôpital's Rule is also referred to as L'Hospital's Rule or Bernoulli's Rule.  
If you want a complete lecture on this topic, we recommend this video.  
Prof Leonard  Evaluating Limits of Indeterminate Forms

Topics You Need To Understand For This Page 

Recommended Books on Amazon (affiliate links)  

L'Hôpital's Rule
If the limit \(\displaystyle{ \lim_{x \to c}{\frac{n(x)}{d(x)}} }\) is indeterminate of the type \(0/0\), then \(\displaystyle{ \lim_{x \to c}{\frac{n(x)}{d(x)}} = \lim_{x \to c}{\frac{n'(x)}{d'(x)}} }\).  
Quick Notes  

This rule can be repeated until a determinate form is found.  
\(x\) can approach a finite value \(c\), \(c^\), \(c^+\), \(\infty\) or \(\infty\)  
Do not use this on determinate forms. It will probably give you the wrong answer.  
For details about determinate and indeterminate forms, click here.  
This is not the quotient rule. For information about the difference, click here.  
Formal Theorem Statement and ProofLet f and g be functions that are differentiable on an open interval \((a,b)\) containing c, except possibly at c itself. Assume that \(g'(x) \ne 0\) for all x in \((a,b)\), except possibly at c itself. If the limit of \(f(x)/g(x)\) as x approaches c produces the indeterminate form \(0/0\), then Here is a video proof of this theorem. Larson Calculus  Proof  L'Hopital's Rule from Larson Texts [3min33secs]
Indeterminate Forms TableDeterminateIndeterminate Forms Table
 
Some Things To Notice  
1. Be careful to use L'Hôpital's Rule only on limits of indeterminate form. If you use it on a determinate form, you may (and probably will) get an incorrect answer. Difference Between L'Hôpital's Rule and the Quotient RuleIt is very easy to confuse L'Hôpital's Rule with the quotient rule since they are so similar. Here is a rundown of the two for comparison. 1. Quotient Rule  The quotient rule is used on a quotient or ratio of terms to calculate the derivative of a function. The result of the quotient rule is the slope of the original function at all points along the curve. So you can use the result to determine the slope, calculate the equation of a tangent line, find extrema and continue on to determine the second derivative, to name a few uses. 2. L'Hôpital's Rule  L'Hôpital's Rule is used only when we want to calculate a limit and can be used only under very specific circumstances (see the discussion below for a complete explanation). L'Hôpital's Rule works like this.
Study Note  When you are learning two similar concepts or you are learning a new concept that is similar to one you already know and you are getting confused, try the technique we used above (setting them sidebyside and comparing and contrasting them). This will help you separate them in your mind and know when to use which technique. This also works for learning similar words in foreign languages. Click here for more study techniques. 
Types of Indeterminate Forms
Notice in the L'Hôpital's Rule equation, we require the limit equation to be in a fraction. This is a strict requirement that cannot be broken. If the limit equation is not a fraction, we cannot use L'Hôpital's Rule. This section contains a discussion of the four main types of equations that you will run across where you need to determine if the limit is an indeterminate form. (This list is found in compact form on the indeterminate forms page. )
Type 1   Indeterminate Quotients
Type 1   Indeterminate Quotients 
Try these practice problems.
Type 1 Practice
Evaluate these limits using L'Hôpital's Rule (if valid), giving your answers in exact form.
Basic 

\(\displaystyle{\lim_{x\to\pi/2}{\left[\frac{\tan(2x)}{x\pi/2}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\pi/2}{\left[\frac{\tan(2x)}{x\pi/2}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to\pi/2}{\left[\frac{\tan(2x)}{x\pi/2}\right]}=2}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\pi/2}{\left[\frac{\tan(2x)}{x\pi/2}\right]}}\)
Solution
If we plug in \( \pi/2 \) for x, we get \( 0/0 \) which is indeterminate. This means we can use L'HÃ´pital's Rule.
\(\displaystyle{ \lim_{x \to \pi/2}{\left[ \frac{\tan(2x)}{x\pi/2} \right]} = }\) \(\displaystyle{\lim_{x \to \pi/2}{\left[ \frac{2 \sec^2(2x)}{1} \right]} = 2 \sec^2(\pi) = 2 }\)
Note: You could also solve this using trig identities (but that's a lot more work).
Final Answer
\(\displaystyle{\lim_{x\to\pi/2}{\left[\frac{\tan(2x)}{x\pi/2}\right]}=2}\)
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\(\displaystyle{\lim_{x\to2}{\left[\frac{3x^2x10}{x^24}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to2}{\left[\frac{3x^2x10}{x^24}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to2}{\left[\frac{3x^2x10}{x^24}\right]}=11/4}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to2}{\left[\frac{3x^2x10}{x^24}\right]}}\)
Solution
First of all, you always want to try direct substitution. When you do this, you get \(0/0\) which is indeterminate.
There are at least two ways to solve this.
We will solve it using L'Hôpital's Rule here. See the finite limits page for an alternative solution.
\(\displaystyle{ \lim_{x \to 2} \left[ \frac{3x^2x10}{x^24} \right] = }\) \(\displaystyle{ \lim_{x \to 2} \left[ \frac{6x1}{2x} \right] = }\) \(\displaystyle{ \frac{6(2)1}{2(2)} = 11/4 }\)
Remember: You can use L'HÃ´pital's Rule only if you have an indeterminate form.
Final Answer
\(\displaystyle{\lim_{x\to2}{\left[\frac{3x^2x10}{x^24}\right]}=11/4}\)
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\(\displaystyle{\lim_{x\to3}{\left[\frac{x^481}{2x^25x3}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to3}{\left[\frac{x^481}{2x^25x3}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to3}{\left[\frac{x^481}{2x^25x3}\right]}=108/7}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to3}{\left[\frac{x^481}{2x^25x3}\right]}}\)
Solution
First of all, you always want to try direct substitution. When you do this, you get \(0/0\) which is indeterminate.
There are at least two ways to solve this.
We will solve it using L'Hôpital's Rule here. See the finite limits page for an alternative solution.
\(\displaystyle{ \lim_{x \to 3}{ \left[\frac{x^481}{2x^25x3}\right] } = }\) \(\displaystyle{ \lim_{x \to 3}{ \left[\frac{4x^3}{4x5}\right] } = \frac{4(3)^3}{4(3)5} = \frac{108}{7} }\)
Remember: You can use L'Hôpital's Rule only if you have an indeterminate form.
Final Answer
\(\displaystyle{\lim_{x\to3}{\left[\frac{x^481}{2x^25x3}\right]}=108/7}\)
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\(\displaystyle{\lim_{x\to27}{\left[\frac{x27}{x^{1/3}3}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to27}{\left[\frac{x27}{x^{1/3}3}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to27}{\left[\frac{x27}{x^{1/3}3}\right]}=27}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to27}{\left[\frac{x27}{x^{1/3}3}\right]}}\)
Solution
Of course, the first thing we want to try is to plug in 27 directly for x. This gives us \(0/0\) which is indeterminate. So we can use L'Hôpital's Rule.
\(\begin{array}{rcl} & & \lim_{x \to 27}{\left[ \frac{x  27}{x^{1/3}  3}\right]} \\ & = & \lim_{x \to 27}{ \frac{1}{(1/3)x^{2/3}} } \\ & = & \lim_{x \to 27}{ \frac{x^{2/3}}{1/3} } \\ & = & \lim_{x \to 27}{ 3x^{2/3} } \\ & = & 3(27)^{2/3} = 3(9) = 27 \end{array}\)
Note   We could have used algebra and factoring to solve this problem. See that solution on the finite limits page.
Final Answer
\(\displaystyle{\lim_{x\to27}{\left[\frac{x27}{x^{1/3}3}\right]}=27}\)
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\(\displaystyle{\lim_{x\to1}{\left[\frac{x^{1/3}1}{x^{1/4}1}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to1}{\left[\frac{x^{1/3}1}{x^{1/4}1}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to1}{\left[\frac{x^{1/3}1}{x^{1/4}1}\right]}=4/3}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to1}{\left[\frac{x^{1/3}1}{x^{1/4}1}\right]}}\)
Solution
Direct substitution yields \(0/0\), which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{ \lim_{x \to 1}{\left[ \frac{x^{1/3}1}{x^{1/4}1}\right] } }\) 
\(\displaystyle{ \lim_{x \to 1}{ \frac{(1/3)x^{2/3}}{(1/4)x^{3/4}} } }\) 
\(\displaystyle{ \lim_{x \to 1}{ \frac{4}{3} \frac{x^{3/4}}{x^{2/3}} } }\) 
\(\displaystyle{ \lim_{x \to 1}{ \frac{4}{3} x^{3/4  2/3} } }\) 
\(\displaystyle{ \lim_{x \to 1}{ \frac{4}{3} x^{9 / 12  8 / 12} } }\) 
\(\displaystyle{ \lim_{x \to 1}{ \frac{4}{3} x^{1 / 12} } }\) 
\(\displaystyle{ \frac{4}{3} 1^{1 / 12} = \frac{4}{3} }\) 
Note   We could have used algebra and factoring to solve this problem. See that solution on the finite limits page.
Final Answer
\(\displaystyle{\lim_{x\to1}{\left[\frac{x^{1/3}1}{x^{1/4}1}\right]}=4/3}\)
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\(\displaystyle{\lim_{x\to0}{\left[\frac{x+\tan(x)}{\sin(x)}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\left[\frac{x+\tan(x)}{\sin(x)}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to0}{\left[\frac{x+\tan(x)}{\sin(x)}\right]}=2}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\left[\frac{x+\tan(x)}{\sin(x)}\right]}}\)
Solution
Direct substitution yields \(0/0\), which is indeterminate. So we can use L'Hôpital's Rule.
\(\begin{array}{rcl} & & \displaystyle{ \lim_{x \to 0}{\left[\frac{x+\tan(x)}{\sin(x)}\right]} } \\ & = & \displaystyle{ \lim_{x \to 0}{\left[\frac{1+\sec^2(x)}{\cos(x)}\right]} } \\ & = & \displaystyle{ \lim_{x \to 0}{\left[\frac{1+[1/\cos^2(x)]}{\cos(x)}\right]} } \\ & = & \displaystyle{ \lim_{x \to 0}{\left[\frac{\cos^2(x)+1}{\cos^3(x)}\right]} } \\ & = & \frac{1+1}{1} = 2 \end{array}\)
After applying L'Hôpital's Rule once, the rest is just trig and algebra.
Final Answer
\(\displaystyle{\lim_{x\to0}{\left[\frac{x+\tan(x)}{\sin(x)}\right]}=2}\)
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\(\displaystyle{\lim_{z\to\pi}{\left[\frac{z\pi}{\sin(z)}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{z\to\pi}{\left[\frac{z\pi}{\sin(z)}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{z\to\pi}{\left[\frac{z\pi}{\sin(z)}\right]}=1}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{z\to\pi}{\left[\frac{z\pi}{\sin(z)}\right]}}\)
Solution
Direct substitution yields \(0/0\), which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{ \lim_{z \to \pi}{\left[ \frac{z  \pi}{\sin(z)} \right]} = }\) \(\displaystyle{ \lim_{z \to \pi}{\left[ \frac{1}{\cos(z)} \right]} = }\) \(\displaystyle{ \frac{1}{1} = 1 }\)
Final Answer
\(\displaystyle{\lim_{z\to\pi}{\left[\frac{z\pi}{\sin(z)}\right]}=1}\)
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\(\displaystyle{\lim_{y\to0}{\left[\frac{\sin(3y)}{\sin(4y)}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{y\to0}{\left[\frac{\sin(3y)}{\sin(4y)}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{y\to0}{\left[\frac{\sin(3y)}{\sin(4y)}\right]}=3/4}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{y\to0}{\left[\frac{\sin(3y)}{\sin(4y)}\right]}}\)
Solution
Direct substitution yields \(0/0\), which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{\lim_{y \to 0}{\left[ \frac{\sin(3y)}{\sin(4y)} \right]} = }\) \(\displaystyle{ \lim_{y \to 0}{\left[ \frac{3\cos(3y)}{4\cos(4y)} \right]} = \frac{3(1)}{4(1)} = 3/4 }\)
Note   There is another way to work this using the identity \(\displaystyle{ \lim_{x \to 0}{\left[ \frac{\sin(x)}{x} \right]} = 1 }\).
Final Answer
\(\displaystyle{\lim_{y\to0}{\left[\frac{\sin(3y)}{\sin(4y)}\right]}=3/4}\)
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\(\displaystyle{\lim_{x\to\infty}{\left[\frac{x+5}{3x+7}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{x+5}{3x+7}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to\infty}{\left[\frac{x+5}{3x+7}\right]}=1/3}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{x+5}{3x+7}\right]}}\)
Solution
Direct substitution yields
\(\displaystyle{ \lim_{x \to \infty}{\left[ \frac{x+5}{3x+7} \right]} = \frac{\infty}{\infty}}\)
This is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{\lim_{x \to \infty}{\left[ \frac{x+5}{3x+7} \right]} = \lim_{x \to \infty}{ \frac{1}{3} } = 1/3}\)
Note   This is the easiest way to work this problem. If you don't know L'Hôpital's Rule, you can use algebra. You will find that solution on the infinite limits page.
Final Answer
\(\displaystyle{\lim_{x\to\infty}{\left[\frac{x+5}{3x+7}\right]}=1/3}\)
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\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^3+x+12}{2x^35x}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^3+x+12}{2x^35x}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^3+x+12}{2x^35x}\right]}=7/2}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^3+x+12}{2x^35x}\right]}}\)
Solution
Direct substitution yields \( \infty  \infty \) in the denominator, which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{\lim_{x \to \infty}{\left[ \frac{7x^3 + x + 12}{2x^3  5x} \right]} = \lim_{x \to \infty}{ \frac{21x^2 + 1}{6x^2} } }\)
Direct substitution still gives us an indeterminate form, \(\infty / \infty\). So we can use L'Hôpital's Rule again.
\(\displaystyle{\lim_{x \to \infty}{ \frac{21x^2 + 1}{6x^2} } = }\) \(\displaystyle{\lim_{x \to \infty}{ \frac{42x}{12x} } = \frac{42}{12} = 7/2}\)
Note   This is the easiest way to work this problem. If you don't know L'Hôpital's Rule, you can use algebra. You will find that solution on the infinite limits page.
Final Answer
\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^3+x+12}{2x^35x}\right]}=7/2}\)
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\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^23x+12}{x^3+4x+127}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^23x+12}{x^3+4x+127}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^23x+12}{x^3+4x+127}\right]}=0}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^23x+12}{x^3+4x+127}\right]}}\)
Solution
Direct substitution yields \(\displaystyle{ \frac{\infty  \infty + 12}{\infty + \infty + 127} }\) which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{\lim_{x \to \infty}{\left[ \frac{14x  3}{3x^2 + 4} \right]}}\)
Again, after direct substitution, we get \( \infty / \infty \), which is also indeterminate. Use L'Hôpital's Rule again.
\(\displaystyle{\lim_{x \to \infty}{\left[ \frac{14x  3}{3x^2 + 4} \right]} = }\) \(\displaystyle{\lim_{x \to \infty}{ \frac{14}{6x} } = \frac{14}{\infty} = 0}\)
Note   This is the easiest way to work this problem. If you don't know L'Hôpital's Rule, you can use algebra. You will find that solution on the infinite limits page.
Final Answer
\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^23x+12}{x^3+4x+127}\right]}=0}\)
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\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^2+x+21}{11x}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^2+x+21}{11x}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^2+x+21}{11x}\right]}=+\infty}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^2+x+21}{11x}\right]}}\)
Solution
Direct substitution yields \(\infty / \infty\) which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{\lim_{x \to \infty}{\left[ \frac{7x^2 + x + 21}{11x} \right]} = }\) \(\displaystyle{\lim_{x \to \infty}{ \frac{14x + 1}{1} }}\)
Now, let's try direct substitution again.
\(\displaystyle{\lim_{x \to \infty}{ \frac{14x + 1}{1} } = \frac{\infty + 1}{1} = \infty}\)
Note   This is the easiest way to work this problem. If you don't know L'Hôpital's Rule, you can use algebra. You will find that solution on the infinite limits page.
Final Answer
\(\displaystyle{\lim_{x\to\infty}{\left[\frac{7x^2+x+21}{11x}\right]}=+\infty}\)
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\(\displaystyle{\lim_{x\to0}{\frac{\sin(x)}{x}}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\frac{\sin(x)}{x}}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to\infty}{\frac{\ln(x)}{x}}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\frac{\ln(x)}{x}}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to0^+}{\frac{\ln(x)}{x}}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0^+}{\frac{\ln(x)}{x}}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to\infty}{\frac{x}{\ln(1+2e^x)}}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\frac{x}{\ln(1+2e^x)}}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to 0}{\frac{\sin(x)}{x+x^2}}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to 0}{\frac{\sin(x)}{x+x^2}}}\)
Final Answer 

\(\displaystyle{\lim_{x\to 0}{\frac{\sin(x)}{x+x^2}}=1}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to 0}{\frac{\sin(x)}{x+x^2}}}\)
Solution
video by Krista King Math 

Final Answer
\(\displaystyle{\lim_{x\to 0}{\frac{\sin(x)}{x+x^2}}=1}\)
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Intermediate 

\(\displaystyle{\lim_{x\to0}{\left[\frac{\tan(x)x}{\sin(x)x}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\left[\frac{\tan(x)x}{\sin(x)x}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to0}{\left[\frac{\tan(x)x}{\sin(x)x}\right]}=2}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\left[\frac{\tan(x)x}{\sin(x)x}\right]}}\)
Solution
Direct substitution yields \(0/0\), which is indeterminate. So we can use L'Hôpital's Rule.
\(\displaystyle{ \lim_{x \to 0}{\left[ \frac{\tan(x)x}{\sin(x)x} \right]} }\) 
\(\displaystyle{ \lim_{x \to 0}{\left[ \frac{\sec^2(x)1}{\cos(x)1} \right]} }\) 
\(\displaystyle{ \lim_{x \to 0}{\left[ \frac{2\sec(x)[\sec(x)\tan(x)]}{\sin(x)} \right]} }\) 
\(\displaystyle{ \lim_{x \to 0}{\left[ \frac{2\sec^2(x)\tan(x)}{\sin(x)} \right]} }\) 
\(\displaystyle{ \lim_{x \to 0}{\left[ \frac{2\sin(x) }{\cos^2(x)\sin(x)\cos(x) } \right]} }\) 
\(\displaystyle{ \lim_{x \to 0}{\left[ \frac{2}{\cos^3(x)} \right]} }\) 
\(\displaystyle{ \frac{2}{1} = 2 }\) 
Final Answer
\(\displaystyle{\lim_{x\to0}{\left[\frac{\tan(x)x}{\sin(x)x}\right]}=2}\)
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\(\displaystyle{\lim_{x\to\infty}{\left[\frac{x^5}{e^{5x}}\right]}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{x^5}{e^{5x}}\right]}}\)
Final Answer 

\(\displaystyle{\lim_{x\to\infty}{\left[\frac{x^5}{e^{5x}}\right]}=0}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{\left[\frac{x^5}{e^{5x}}\right]}}\)
Solution
If we try direct substitution first, we get \( \infty / \infty \) which is indeterminate. So we can use L'Hôpital's Rule.
\(\begin{array}{rcl} & & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{x^5}{e^{5x}} \right]} } \\ & = & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{5x^4}{5e^{5x}} \right]} } \\ & = & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{x^4}{e^{5x}} \right]} } \\ & = & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{4x^3}{5e^{5x}} \right]} } \\ & = & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{12x^2}{25e^{5x}} \right]} } \\ & = & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{24x}{125e^{5x}} \right]} } \\ & = & \displaystyle{ \lim_{x \to \infty}{\left[ \frac{24}{625e^{5x}} \right]} } \\ & = & \displaystyle{ \frac{24}{\infty} = 0 } \end{array}\)
Note   In this solution we used L'Hôpital's Rule five times until we no longer had an indeterminate form. Not shown here is the fact that, before each use of L'Hôpital's Rule, we tested for indeterminate forms by direct substitution.
Final Answer
\(\displaystyle{\lim_{x\to\infty}{\left[\frac{x^5}{e^{5x}}\right]}=0}\)
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\(\displaystyle{ \lim_{x \to 0^}{ \frac{e^{1/x}}{x} } }\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{ \lim_{x \to 0^}{ \frac{e^{1/x}}{x} } }\)
Hint 

Use the substitution \(u=1/x\) and move the exponential to the denominator.
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{ \lim_{x \to 0^}{ \frac{e^{1/x}}{x} } }\)
Final Answer 

\(\displaystyle{ \lim_{x \to 0^}{ \frac{e^{1/x}}{x} } = 0 }\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{ \lim_{x \to 0^}{ \frac{e^{1/x}}{x} } }\)
Hint
Use the substitution \(u=1/x\) and move the exponential to the denominator.
Solution
Direct substitution yields \(0/0\) which is indeterminate. So our next logical step is to use L'HÃ´pital's Rule. However, when we do that, we get an even more complicated result.
If we use the hint, the limit will approach \(\infty\) and we have \(\displaystyle{ \lim_{u \to \infty}{ \frac{e^{u}}{1/u} } = \lim_{u \to \infty}{ \frac{u}{e^{u}} } }\).
Now we can use L'HÃ´pital's Rule (since direct substitution yields the indeterminate fraction \(\infty / \infty\)) to get
\(\displaystyle{ \lim_{u \to \infty}{ \frac{u}{e^{u}} } = }\) \(\displaystyle{ \lim_{u \to \infty}{ \frac{1}{e^{u}} } = }\) \(\displaystyle{ \frac{1}{e^{\infty}} = 0 }\)
Final Answer
\(\displaystyle{ \lim_{x \to 0^}{ \frac{e^{1/x}}{x} } = 0 }\)
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Type 2   Indeterminate Products
Type 2   Indeterminate Products 
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Type 2 Practice
Evaluate these limits using L'Hôpital's Rule (if valid), giving your answers in exact form.
\(\displaystyle{\lim_{x\to0^+}{\sqrt{x}\ln(x)}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0^+}{\sqrt{x}\ln(x)}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to0}{\cot(2x)\sin(6x)}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\cot(2x)\sin(6x)}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{ \lim_{x \to \pm \infty}{ x \arctan\left[ \frac{1}{2x^2} \right] } }\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{ \lim_{x \to \pm \infty}{ x \arctan\left[ \frac{1}{2x^2} \right] } }\)
Final Answer 

\( 0 \)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{ \lim_{x \to \pm \infty}{ x \arctan\left[ \frac{1}{2x^2} \right] } }\)
Solution
This limit is part of a improper integral problem. You do not need to know how to work improper integrals to understand how to evaluate this limit.
video by Michael Penn 

Final Answer
\( 0 \)
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Type 3   Indeterminate Differences
Type 3   Indeterminate Differences 
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Type 3 Practice
Evaluate these limits using L'Hôpital's Rule (if valid), giving your answers in exact form.
\(\displaystyle{\lim_{x\to\infty}{(x^2x)}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{(x^2x)}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to\infty}{(xe^{1/x}x)}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{(xe^{1/x}x)}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to1}{\left[\frac{1}{\ln(x)}\frac{1}{x1}\right]}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to1}{\left[\frac{1}{\ln(x)}\frac{1}{x1}\right]}}\)
Solution
video by PatrickJMT 

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Type 4   Indeterminate Powers
Type 4   Indeterminate Powers 
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Type 4 Practice
Evaluate these limits using L'Hôpital's Rule (if valid), giving your answers in exact form.
\(\displaystyle{ \lim_{x \to 0^+}{ x^x } }\)
Problem Statement
Evaluate the limit \(\displaystyle{ \lim_{x \to 0^+}{ x^x } }\), giving your answer in exact terms.
Solution
video by BriTheMathGuy 

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\(\displaystyle{\lim_{n\to\infty}{n^{2/n}}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{n^{2/n}}}\)
Final Answer 

\(\displaystyle{\lim_{n\to\infty}{n^{2/n}}=1}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{n^{2/n}}}\)
Solution
Direct substitution yields \( \infty^0 \) which is indeterminate. We can't directly use L'Hôpital's Rule because L'Hôpital's Rule requires a fraction. So we have to use a trick.
Let's call the limit, \(y\), i.e. \(\displaystyle{ y = \lim_{n \to \infty}{n^{2/n}} }\) and take the natural log of both sides. This yields
\(\begin{array}{rcl} \ln(y) & = & \displaystyle{ \lim_{n \to \infty}{\ln(n^{2/n})} } \\ & = & \displaystyle{ \lim_{n \to \infty}{\frac{2\ln(n)}{n}} } \\ & = & \displaystyle{ \lim_{n \to \infty}{\frac{2/n}{1}} = 0 } \\ \ln(y) & = & 0 \\ e^{\ln(y)} & = & e^0 \\ y & = & 1 \end{array}\)
Final Answer
\(\displaystyle{\lim_{n\to\infty}{n^{2/n}}=1}\)
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\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{k}}}\) where \(k\) is a positive constant
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{k}}}\) where \(k\) is a positive constant
Final Answer 

\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{k}}=1}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{k}}}\) where \(k\) is a positive constant
Solution
\(\displaystyle{ y = \lim_{n \to \infty}{ \sqrt[n]{k} } = \lim_{n \to \infty}{ k^{1/n} } }\) 
\(\displaystyle{ \ln(y) = \ln\left[ \lim_{n \to \infty}{ k^{1/n} } \right] }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \ln\left( k^{1/n} \right) } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ (1/n)\ln(k) } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{\ln(k)}{n} } }\) 
\( \ln(y) = 0 \) 
\( e^{\ln(y)} = e^0 \) 
\( y = 1 \) 
Final Answer
\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{k}}=1}\)
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\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{n^k}}}\) where \(k\) is a positive constant
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{n^k}}}\) where \(k\) is a positive constant
Final Answer 

\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{n^k}}=1}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{n^k}}}\) where \(k\) is a positive constant
Solution
\(\displaystyle{ y = \lim_{n \to \infty}{ \sqrt[n]{n^k} } = \lim_{n \to \infty}{ n^{k/n} } }\) 
\(\displaystyle{ \ln(y) = \ln \left[ \lim_{n \to \infty}{ n^{k/n} } \right] }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \ln \left[ n^{k/n} \right] } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ (k/n)\ln(n) } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{k\ln(n)}{n} } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{k(1/n)}{1} } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{k}{n} } }\) 
\( \ln(y) = 0 \) 
\( e^{\ln(y)} = e^0 \) 
\( y = 1 \) 
Final Answer
\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{n^k}}=1}\)
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\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{\ln(n)}}}\)
Problem Statement 

Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{\ln(n)}}}\)
Final Answer 

\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{\ln(n)}}=1}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{\ln(n)}}}\)
Solution
\(\displaystyle{ y = \lim_{n \to \infty}{ \sqrt[n]{\ln(n)} } = \lim_{n \to \infty}{ (\ln(n))^{1/n} } }\) 
\(\displaystyle{ \ln(y) = \ln \left[ \lim_{n \to \infty}{(\ln(n))^{1/n} } \right] }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \ln \left[ (\ln(n))^{1/n} \right] } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ (1/n)\ln(\ln(n)) } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{\ln(\ln(n))}{n} } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{1/n}{\ln(n)(1)} } }\) 
\(\displaystyle{ \ln(y) = \lim_{n \to \infty}{ \frac{1}{n\ln(n)} } }\) 
\( e^{\ln(y)} = e^0 \) 
\(y = 1\) 
Final Answer
\(\displaystyle{\lim_{n\to\infty}{\sqrt[n]{\ln(n)}}=1}\)
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\(\displaystyle{\lim_{x\to\infty}{(e^x+x)^{1/x}}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to\infty}{(e^x+x)^{1/x}}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{\lim_{x\to0}{(\cos(3x))^{5/x}}}\)
Problem Statement
Evaluate this limit using L'Hopital's Rule (if valid), giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{(\cos(3x))^{5/x}}}\)
Solution
video by PatrickJMT 

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\(\displaystyle{ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } }\)
Problem Statement 

Evaluate this limit, giving your answer in exact terms. \(\displaystyle{ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } }\).
Hint 

Use natural logarithms to get the \(x\) out of the exponent.
Problem Statement 

Evaluate this limit, giving your answer in exact terms. \(\displaystyle{ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } }\).
Final Answer 

\(\displaystyle{ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } = \frac{1}{e} }\)
Problem Statement
Evaluate this limit, giving your answer in exact terms. \(\displaystyle{ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } }\).
Hint
Use natural logarithms to get the \(x\) out of the exponent.
Solution
First let's assign the limit to the variable \(y\). 
\(\displaystyle{ y = \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } }\) 
Use the hint by taking the natural logarithm of both sides of the equation. 
\(\displaystyle{ \ln(y) = \ln \left[ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } \right] }\) 
On the left, move the natural logarithm inside of the limit. We can do this since the natural log function is continuous. 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \ln \left( \frac{x}{x+1} \right)^x } }\) 
Now we can move the exponent out in front using the basic logarithm law \(\ln(x^y) = y\ln x\) 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ x \ln \left( \frac{x}{x+1} \right) } }\) 
In order to use L'Hopital's Rule, we need \(0/0\) or \(\infty / \infty\). Let's move the \(x\) out in front to the denominator. 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{ \ln (x/(x+1)) }{1/x} } }\) 
Now we have a limit that goes to \(0/0\). (If you are not sure about the top going to zero, convince yourself by working it on the side. One way is to multiply the numerator and denominator by \(1/x\) and taking the limit of numerator and denominator separately.) So now we can apply L'Hopital's Rule. 
Let's take the derivatives of the numerator and denominator individually. First the numerator. 
\(\displaystyle{ \frac{d}{dx} \ln (x/(x+1)) }\) 
Let's break the natural logarithm into two terms to make the derivative easier to evaluate. 
\(\displaystyle{ \frac{d}{dx} [ \ln(x)  \ln(x+1) ] }\) 
Now take the derivative. 
\(\displaystyle{ \frac{1}{x}  \frac{1}{x+1} = \frac{x+1x}{x(x+1)} = \frac{1}{x(x+1)} }\) 
Now evaluate the derivative of the denominator. 
\(\displaystyle{ \frac{d[1/x]}{dx} = \frac{d[x^{1}]}{dx} = x^{2} }\) 
Now let's put the results of the derivatives back into the limit to see what we have. 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{1}{x(x+1)} \left[ \frac{1}{x^{2}} \right] } }\) 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{1}{x(x+1)} (x^2) } }\) 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{x^2}{x(x+1)} } }\) 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{x}{x+1} } }\) 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{x(1/x)}{(x+1)(1/x)} } }\) 
\(\displaystyle{ \ln(y) = \lim_{x \to \infty}{ \frac{1}{1+1/x} } }\) 
Taking the limit gives us \(1/(1+0) = 1\) 
So our equation is now \( \ln(y) = 1 \) 
Now we need to solve for \(y\) which is the original limit. 
\( e^{\ln y} = e^{1} \to y = 1/e\) 
So, a question that you may have is, why did we assign the original limit to the variable \(y\) when we didn't really use it except at the very end of the problem? Couldn't we have just left it off? Yes, we could have done without it. However, we have found that it is very easy to forget to convert back at the end of the problem and just say the limit is \(1\) in this case. So we use it as a placeholder and as a reminder that we have that last step before we get our final answer. Notice that we carry it along in the problem, so that we don't forget it. This is important too.
Of course, check with your instructor to see what they expect.
Final Answer
\(\displaystyle{ \lim_{x \to \infty}{ \left( \frac{x}{x+1} \right)^x } = \frac{1}{e} }\)
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How It Works
Okay, now that you know something about L'Hôpital's Rule, under what conditions you can use it and various situations you will encounter, let's watch a video to help explain how it works and see some examples. This video is indepth and a bit long but is well worth watching. The instructor is easy to listen to and he does a good job of explaining the details of L'Hôpital's Rule as well as giving examples and supporting arguments.
video by MIT OCW 

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