Squeeze Theorem
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Alternate Names For The Squeeze Theorem |
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The Sandwich Theorem |
The Pinching Theorem |
Squeeze 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 Squeeze Theorem is a powerful theorem that allows us to prove several important limits, including this 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. 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.
This video explains the squeeze theorem with a graph and it is a good place to start.
Here is a great video showing the proof of \(\displaystyle{ \lim_{x \to 0}{\left[ xsin(1/x)\right]} = 0 }\) using the squeeze theorem (which he calls the pinching theorem). He does some unusual things here, so it is important to watch this video to see this technique.
Squeeze Theorem At Infinity |
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In the above discussion, we used the squeeze 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.
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