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Calculus presents us with various useful computational tricks, especially in the field of limits. When faced with oscillating functions or functions with undefined points, taking the limit can become a difficult task. Luckily, The Squeeze, or Sandwich, Theorem is just the trick for dealing with tricky functions such as these.
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Team The Squeeze Theorem Teachers
The Squeeze Theorem is a limit evaluation method where we "squeeze" an indeterminate limit between two simpler ones. The "squeezed" or "bounded" function approaches the same limit as the other two functions surrounding it.
More precisely, the Squeeze Theorem states that for functions \(f\), \(g\), and \(h\) such that:
\[g(x) \leq f(x) \leq h(x)\]
if\[lim_{x \rightarrow A } g(x)= lim_{x \rightarrow A } h(x)=L\]
for a constant \(L\), then:
\[lim_{x \rightarrow A} f(x)=L\]
Fig. 1. As \(f(x)\) is "squeezed" between \(g(x)\) and \(h(x)\), the Squeeze Theorem can be applied to evaluate the limit of \(f(x)\) at \(x = 0\).
Simply put, \(f(x)\) is "squeezed" between \(g(x)\) and \(h(x)\). As \(g(x)\) and \(h(x)\) are equal at the point A such that \(g(A)=h(A)=L\), then \(f(A)=L\) , as there is no room between the other two functions for \(f\) to take on any other value.
We will assume that
Upon these assumptions, you want to prove that:
\[lim_{x \rightarrow A} f(x)=L\]
See the image below for a visual explanation of the variables!
Let an arbitrary epsilon such that \(\epsilon > 0\) be known. To prove the Squeeze Theorem, we must find a delta \(\delta > 0\) such that \(|f(x)-L|< \epsilon\) whenever \(0< |x-A|< \delta\) where L is the evaluation of the limit as \(x\) approaches the point \(A\).
Now \(\lim_{x \rightarrow A} g(x)=L\) by definition, so there must exist some \(\delta_g > 0\) such that:
\(|g(x)-L|< \epsilon\) for all \(0 < |x-A|< \delta_g\):
Using absolute value laws
\[-\epsilon + L < g(x) < \epsilon + L\]
for all
\[0<|x-A|<\delta_g\]
Similarly, \(lim_{x \rightarrow A} h(x)=L\) by definition, so there must exist some \(\delta_g > 0\)such that
\(|h(x)-L|< \epsilon\) for all \(0 < |x-A|< \delta_g\).
Using absolute value laws
\[- \epsilon + L < h(x)< \epsilon + L \]
for all
\[0<|x-A|<\delta_g\]
Fig. 2. Visual explanation of the geometric derivation of (1) and (2).
Since \(g(x)\leq f(x) \leq h(x)\) for all \(x\) on some open interval containing \(A\), there must exist some \(\delta_f > 0\) such that
(3) \(g(x) \leq f(x) \leq h(x)\) for all \(0< |x-A|< \delta_f\)
Where \(\delta = min (\delta_g, \delta_h, \delta_f)\), then by (1), (2), and (3)
\[-\epsilon + L < g(x) \leq f(x) \leq h(x) < L + \epsilon\] for all \(0<|x-A|<\delta\)
Thus
\(-\epsilon< f(x)-L < \epsilon \) for all/ \(0<|x-A|<\delta\)
Using absolute value laws
\( |f(x)-L| < \epsilon\) for all \(0<|x-A|<\delta\)
Then, by definition:
\[lim_{x \rightarrow A} f(x)=L\]
The Squeeze Theorem should be used as a last resort. When solving limits, one should always try to solve through algebraic or simple manipulation first. If algebra fails, the Squeeze Theorem may be a viable option for limit solving.
Indeed, to calculate \(lim_{x \rightarrow A} f(x)\), we must first find two functions \(g(x)\) and \(h(x)\) that bound \(f(x)\) and such that:
\[lim_{x \rightarrow A}g(x)=lim_{x \rightarrow A} h(x)\]
The Squeeze Theorem cannot be applied if the limits of the bounding functions are not equal.
Let's start with a simple example!
Use the Squeeze Theorem to evaluate
\[lim_{x \rightarrow 0} x^2 \cos \left( \dfrac{1}{x^2} \right)\]
When we plug in \(x = 0\), we are met with an undefined form \(cos \left( \frac{1}{0} \right)\). This is a perfect candidate for the Squeeze Theorem!This is an example of a general scenario: the Squeeze Theorem can be applied to find the limit of trigonometric functions damped by polynomial terms.
It is essential to know that the range of \(\cos\) (anything) and \(\sin\) (anything) will always be \([-1, 1]\) (as long as it is not translated up/down or vertically stretched/compressed)!
Fig. 3. Example 1 graph.
Now, let's try something a bit more complex.
Find \(lim_{x \rightarrow - \infty} \dfrac{7x^2-\sin(5x)}{x^2+15}\)
When we plug in \(- \infty\), we are left with the indeterminate form \(\dfrac{\infty}{\infty}\). Again, since a trigonometric function appears, this is a perfect candidate for the Squeeze Theorem!Following the same strategy as before, start with the trigonometric function \(f(x)=\sin(5x)\), and build up to
\[f(x)=\dfrac{7x^2-\sin(5x)}{x^2+15}\]
Let's take a look!
Fig. 4. Example 2 graph.
for a constant \(L\), then: \[lim_{x \rightarrow A} f(x)=L\]
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What is the squeeze theorem?
The Squeeze Theorem is a method for solving limits that cannot be solved through algebra or other simple manipulations.
How to evaluate a limit using the squeeze theorem?
To solve with the Squeeze Theorem:
How to apply the squeeze theorem?
To solve with the Squeeze Theorem:
When does the squeeze theorem fail?
The Squeeze Theorem cannot be applied if two-sided limit does not exist. In other words, if the right-hand and left-hand limits are not equal, the Squeeze Theorem will not work.
How do you solve the Squeeze Theorem?
To solve with the Squeeze Theorem:
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