17Calculus Limits - Pinching Theorem
The Pinching Theorem is also referred to as The Sandwich Theorem or The Squeeze Theorem.
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Pinching Theorem |
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For an interval I containing a point a, if we have three functions, \(f(x), g(x), h(x)\) all three defined on the interval I, except at a and we have \(\displaystyle{ g(x) \leq f(x) \leq h(x) }\) for all points in I except possibly at \(x=a\) and we know that \(\displaystyle{ \lim_{x \to a}{g(x)} = \lim_{x \to a}{h(x)} = L }\) Then \(\displaystyle{ \lim_{x \to a}{f(x)} = L }\) |
The Pinching Theorem is a powerful theorem that allows us to determine several important limits, including this important trig limit.
\(\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 }\) |
To help you get your head around this theorem, here is a graph that intuitively shows you the idea of the proof of the above limit. (See this separate page for two additional proofs of this limit.) The red line is the function
\( \color{#BF003F}{ f(x) = \sin(x)/x }\)
, the green line is \( \color{#007F7F}{g(x) = \cos(x) }\) and the blue line is \(\color{#0000FF}{h(x) = 1}\). Notice that near, i.e. within an interval, around \(x=0\),
\( \color{#007F7F}{g(x)} \leq \color{#BF003F}{f(x)} \leq \color{#0000FF}{h(x)}\) and
\(\displaystyle{ \color{#007F7F}{\lim_{x \to 0}{g(x)} = 1} }\) |
\(\displaystyle{ \color{#0000FF}{\lim_{x \to 0}{h(x)} = 1} }\) |
which means \(\displaystyle{ \color{#BF003F}{\lim_{x \to 0}{f(x)} = 1} }\)
The trick comes in when you have to find two functions \(g(x)\) and \(h(x)\) that satisfy the theorem. However, if you graph \(f(x)\), sometimes you will be able to find a couple of functions that work. However, there is one idea that will sometimes work. If you have the trig functions \(\sin(x)\) or \(\cos(x)\), these functions are always between \(-1\) and \(+1\). So setting up an inequality with those will sometimes give you a place to start.
Okay, let's look at some videos to explain this in more detail. The first video explains the pinching(squeeze) theorem with a graph and it is a good place to start.
PatrickJMT - The Squeeze Theorem for Limits, Example 1 [7min-12secs]
video by PatrickJMT |
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Here is a great video showing the proof of \(\displaystyle{ \lim_{x \to 0}{\left[ x\sin(1/x)\right]} = 0 }\) using the pinching theorem. He does some unusual things here, so it is important to watch this video to see this technique.
Dr Chris Tisdell - Limit of a function: Pinching theorem with streamlined method of solution [4min-22secs]
video by Dr Chris Tisdell |
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Pinching Theorem At Infinity
In the above discussion, we used the pinching theorem when the limit variable was going to a finite number. However, we can apply the same technique when the limit variable is going to infinity, i.e. we have the limit \(\displaystyle{ \lim_{x \to \infty}{f(x)} }\). We just need to show that the function is bounded above and below by two functions and that the limits of both bounding functions goes to the same value as x goes to infinity. You will find some practice problems below demonstrating this technique.
Practice
Unless otherwise instructed, evaluate these limits using the pinching theorem. Give your answers in exact form.
Basic |
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If \(3x \leq f(x) \leq x^3+2\) on \([0,2]\), evaluate \(\displaystyle{ \lim_{x \to 1}{f(x)} }\) using the pinching theorem.
Problem Statement |
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If \(3x \leq f(x) \leq x^3+2\) on \([0,2]\), evaluate \(\displaystyle{ \lim_{x \to 1}{f(x)} }\) using the pinching theorem. Give your answer in exact form.
Solution |
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470 video
video by PatrickJMT |
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If \(4\leq f(x)\leq x^2+6x-3\) for all \(x\), evaluate \(\displaystyle{ \lim_{x \to 1}{f(x)} }\) using the pinching theorem.
Problem Statement |
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If \(4\leq f(x)\leq x^2+6x-3\) for all \(x\), evaluate \(\displaystyle{ \lim_{x \to 1}{f(x)} }\) using the pinching theorem. Give your answer in exact form.
Solution |
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472 video
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Evaluate \(\displaystyle{\lim_{x\to4}{f(x)}}\) if, for all \(x\), \(4x-9\leq f(x)\leq x^2-4x+7\) using the pinching theorem.
Problem Statement |
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Evaluate \(\displaystyle{\lim_{x\to4}{f(x)}}\) if, for all \(x\), \(4x-9\leq f(x)\leq x^2-4x+7\) using the pinching theorem. Give your answer in exact form.
Solution |
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477 video
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\(\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}}\)
Final Answer |
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\(\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}=0}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}}\)
Solution |
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1948 video
Final Answer |
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\(\displaystyle{ \lim_{n\to\infty}{\frac{\cos^2n}{n}}=0}\) |
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Intermediate |
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\(\displaystyle{\lim_{x\to0}{\left[x^2\cdot\cos(1/x^2)\right]}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{x\to0}{\left[x^2\cdot\cos(1/x^2)\right]}}\)
Solution |
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471 video
video by PatrickJMT |
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\(\displaystyle{\lim_{x\to0}{x^4\sin(3/x)}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{x\to0}{x^4\sin(3/x)}}\)
Solution |
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473 video
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\(\displaystyle{\lim_{n\to\infty}{\frac{1}{n^3}\sin(n^2)}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\frac{1}{n^3}\sin(n^2)}}\)
Solution |
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474 video
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\(\displaystyle{\lim_{n\to\infty}{\frac{(-1)^n+n^2}{n^2}}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{\frac{(-1)^n+n^2}{n^2}}}\)
Solution |
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475 video
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\(\displaystyle{\lim_{n\to\infty}{n^{-n^3}}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{n\to\infty}{n^{-n^3}}}\)
Solution |
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476 video
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\(\displaystyle{\lim_{x\to0}{x^2\cos(10x)}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{x\to0}{x^2\cos(10x)}}\)
Solution |
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478 video
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\(\displaystyle{\lim_{x\to0}{x^2\cos(1/\sqrt[3]{x})}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{x\to0}{x^2\cos(1/\sqrt[3]{x})}}\)
Solution |
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479 video
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\(\displaystyle{\lim_{x\to0}{\sqrt[3]{x}\sin(1/x)}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{x\to0}{\sqrt[3]{x}\sin(1/x)}}\)
Solution |
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480 video
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\(\displaystyle{\lim_{x\to0}{x^2\sin(1/x^2)}}\)
Problem Statement |
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Evaluate the limit using the pinching theorem. Give your answer in exact form. \(\displaystyle{\lim_{x\to0}{x^2\sin(1/x^2)}}\)
Solution |
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481 video
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Related Topics and Links
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Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
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Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
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Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
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Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
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Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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Topics Listed Alphabetically
Single Variable Calculus |
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Differential Equations |
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Practice Instructions
Unless otherwise instructed, evaluate these limits using the pinching theorem. Give your answers in exact form.
