## One-Sided Limits

### Video List

DrChrisTisdell Videos

limit of a function

A01  A02  limits with absolute value

A03  A04  finite limits

B01  piecewise function limit

A05  finite limit

A06  one-sided limit

various one-sided limits

A07  one-sided limit

B02  prove a limit does not exist

B03  limit with absolute value

A08  A09  a couple of easy limits

### Search 17calculus

One-sided limits are finite limits where we evaluate the limit from each side individually. The notation we use is

 $$\displaystyle{ \lim_{x \to a^-}{f(x)} }$$ evaluate the limit on the left side of $$a$$, i.e. values of $$x < a$$ $$\displaystyle{ \lim_{x \to a^+}{f(x)} }$$ evaluate the limit on the right side of $$a$$, i.e. values of $$x > a$$

The negative and positive sign that look like exponents on the finite value $$a$$ indicate the side that we are looking at.

One of the reasons we need to look at limits on both sides of some number is when we are determining continuity. As you know from the continuity page, one of the requirements for continuity is that the limit at a point must exist. In order for a limit to exist, the limit from the left must be equal to the limit from the right, i.e.
$$\displaystyle{ \lim_{x \to a^-}{f(x)} = \lim_{x \to a^+}{f(x)} }$$

Notice that we are NOT saying that the function value must be equal to the limit or even that the function need be defined at $$x=a$$, only that the limit be equal on both sides of $$a$$.

Here is a great video to build your intuition of one-sided limits. He uses an absolute value function to discuss the idea of one-sided limits and limits that do not exist.

Here is a good video showing a graph with several one-sided limits.

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### One-Sided Limits Resources

Practice Problems

Instructions - - Unless otherwise instructed, evaluate the following limits. Give your answers in exact form.

 Level A - Basic

Practice A01

$$\displaystyle{ \lim_{x \to 1}{\frac{\left| x-1 \right|}{x-1}} }$$

Practice A02

$$\displaystyle{ \lim_{x \to 0^-}{ \left( \frac{1}{x} - \frac{1}{|x|} \right) } }$$

Practice A03

$$\displaystyle{ \lim_{x \to 5^+}{\frac{6}{x-5}} }$$

Practice A04

$$\displaystyle{ \lim_{x \to 0}{\frac{x-1}{x^2(x+2)}} }$$

##### Practice Problem [ A01 ] Solution

$$\displaystyle{ \lim_{x \to 1}{\frac{\left| x-1 \right|}{x-1}} }$$ does not exist

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##### Practice Problem [ A02 ] Solution

$$\displaystyle{ \lim_{x \to 0^-}{ \left( \frac{1}{x} - \frac{1}{|x|} \right) } = - \infty }$$

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##### Practice Problem [ A03 ] Solution

$$\displaystyle{ \lim_{x \to 5^+}{\frac{6}{x-5}} = +\infty }$$

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##### Practice Problem [ A04 ] Solution

$$\displaystyle{ \lim_{x \to 0}{\frac{x-1}{x^2(x+2)}} = -\infty }$$

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Practice A05

$$\displaystyle{ \lim_{x \to -4^-}{ |x+4| } }$$

Practice A06

$$\displaystyle{ \lim_{x \to 5^+}{ \sqrt{x^2-25} } }$$

Practice A07

$$\displaystyle{ \lim_{x \to 5-}{\sqrt{x(5-x)}} }$$

Practice A08

$$\displaystyle{ \lim_{x \to 0}{1/x} }$$

##### Practice Problem [ A05 ] Solution

$$\displaystyle{ \lim_{x \to -4^-}{ |x+4| } = 0 }$$

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##### Practice Problem [ A06 ] Solution

$$\displaystyle{ \lim_{x \to 5^+}{ \sqrt{x^2-25} } = 0 }$$

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##### Practice Problem [ A07 ] Solution

$$\displaystyle{ \lim_{x \to 5-}{\sqrt{x(5-x)}} = 0 }$$

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##### Practice Problem [ A08 ] Solution

$$\displaystyle{ \lim_{x \to 0}{1/x} }$$ does not exist

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Practice A09

$$\displaystyle{ \lim_{x \to 0}{1/x^2} }$$

##### Practice Problem [ A09 ] Solution

$$\displaystyle{ \lim_{x \to 0}{1/x^2} = +\infty }$$

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 Level B - Intermediate

Practice B01

$$\displaystyle{ f(x) = \left\{ \begin{array}{lr} x+3 & x \leq 1 \\ x^2-2x & x >1 \end{array} \right. }$$
Evaluate $$\displaystyle{ \lim_{x \to 1}{f(x)} }$$

Practice B02

Prove that the limit $$\displaystyle{ \lim_{x \to 0}{\frac{|x|}{x}} }$$ does not exist.

Practice B03

$$\displaystyle{ \lim_{x \to 0}{\frac{x-2|x|}{|x|}} }$$

##### Practice Problem [ B01 ] Solution

$$\displaystyle{ \lim_{x \to 1}{f(x)} }$$ does not exist

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##### Practice Problem [ B03 ] Solution

$$\displaystyle{ \lim_{x \to 0}{\frac{x-2|x|}{|x|}} }$$ does not exist