Calculus Squeeze Theorem

In this section, ask-math explains you the calculus squeeze theorem.
The limit of function that is squeezed between two other functions, each function will have same limit at given value of x. If two functions squeeze together at a particular point, then any function trapped between them will get squeezed to that same point.
1) If two functions sandwich together at a particular point, then any function trapped between them will get squeezed to that same point.
2) The squeeze or sandwich theorem deals with the limit values rather instead of function values.
3) Squeeze theorem or sandwich theorem is also called Pinch theorem.
If h(x) $\leq$ f(x) $\leq$ g(x), for all x in open interval containing 'c' except possibly at 'c' itself, and if

$\lim_{x->c} h(x) = L = \lim_{x->c} g(x)$ then
$\lim_{x->c}f(x)$ exists and is equal to 'L'.

Examples on calculus squeeze theorem

1) Evaluate $\lim_{x->0} x * sin(\frac{1}{x})$

Solution : As we know that the sine function lies between -1 and 1.

$-1 \leq sin(\frac{1}{x})\leq 1$ for all x

So, $-|x| \leq sin(\frac{1}{x})\leq |x|$ for all x

We know that

$\lim_{x->0}|x|$ = 0 and $\lim_{x->0}-|x|$ = 0

So according to squeeze theorem $\lim_{x->0} x * sin(\frac{1}{x})$ = 0

2) Use sandwich theorem to find $\lim_{x->c}f(x)$, c = 0 , $4 - x^{2}\leq f(x)\leq 4 + x^{2}$
Solution : Let h(x) = $4 - x^{2}$ and g(x) = $4 + x^{2}$
Now we will use squeeze theorem to find lim f(x) as x approaches to 0
h(x) $\leq$ f(x) $\leq$ g(x)
$\lim_{x->0}4 - x^{2}$ = 4 and $\lim_{x->0}4 + x^{2}$ = 4

$4 \leq \lim_{x->0}f(x)\leq 4 $

So, $\lim_{x->0}f(x)$ = 4

3) Evaluate $\lim_{x->0} x * cos(\frac{1}{x})$

Solution : As we know that the sine function lies between -1 and 1.

$-1 \leq cos(\frac{1}{x})\leq 1$ for all x

So, $-|x| \leq cos(\frac{1}{x})\leq |x|$ for all x

We know that

$\lim_{x->0}|x|$ = 0 and $\lim_{x->0}-|x|$ = 0

So according to squeeze theorem $\lim_{x->0} x * cos(\frac{1}{x})$ = 0

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