Question

How do you use the squeeze theorem to show that $\displaystyle \lim_{x \to 0}{{x}^{4}}\cos 2x=0$?

Answer 1

We first explain the squeeze theorem. We also explain the process of finding the limit value rather than the function value. Using trigonometric identity, we find the range of $\cos 2x$ and then using that we find the limit value of the function ${{x}^{4}}\cos 2x$.

Complete step by step answer:
We explain the squeeze theorem for finding the limit of $\displaystyle \lim_{x \to 0}{{x}^{4}}\cos 2x=0$.
The theorem tells that if two functions are together to a particular point then any function trapped in between them will be also squeezed to that point. It is more of a limit value decider than the function value. it is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed.
In the case of our given function ${{x}^{4}}\cos 2x$, the limit value of the whole function can’t be found directly. So, we try to find the range of the trigonometric function which in turn helps in finding the limit.
We know that $\forall x\in \mathbb{R}$ the value of $\cos 2x$ will be in between -1 and 1.
This means $-1\le \cos 2x\le 1$.
Multiplying the function ${{x}^{4}}$ with $-1\le \cos 2x\le 1$, we get $-{{x}^{4}}\le {{x}^{4}}\cos 2x\le {{x}^{4}}$.
We get the range value of the given function. Now we can apply limit on the inequality of $-{{x}^{4}}\le {{x}^{4}}\cos 2x\le {{x}^{4}}$
and find the limit value.
$\displaystyle \lim_{x \to 0}\left( -{{x}^{4}} \right)\le \displaystyle \lim_{x \to 0}\left( {{x}^{4}}\cos 2x \right)\le \displaystyle \lim_{x \to 0}\left( {{x}^{4}} \right)$.
We place the value of $x$ and get $0\le \displaystyle \lim_{x \to 0}\left( {{x}^{4}}\cos 2x \right)\le 0$.
Therefore, $\displaystyle \lim_{x \to 0}{{x}^{4}}\cos 2x=0$. Thus, proved.

Note: We find that the limit is same for the exponential function $\displaystyle \lim_{x \to 0}\left( -{{x}^{4}} \right)\le \displaystyle \lim_{x \to 0}\left( {{x}^{4}}\cos 2x \right)\le \displaystyle \lim_{x \to 0}\left( {{x}^{4}} \right)$. Both the limits tend to 0. This theorem is sometimes called sandwich theorem as the value or the range gets decided by this theorem.