Question

Question 5: prove that it’s =0

Answer 1

Answer:

Proof in explanation.

Step-by-step explanation:

I'm going to attempt this by squeeze theorem.

We know that $\cos(\frac{2}{x})$ is a variable number between -1 and 1 (inclusive).

This means that $-1 \le \cos(\frac{2}{x}) \le 1$.

$x^4 \ge 0$ for all value $x$. So if we multiply all sides of our inequality by this, it will not effect the direction of the inequalities.

$-x^4 \le x^4 \cos(\frac{2}{x}) \le x^4$

By squeeze theorem, if  $-x^4 \le x^4 \cos(\frac{2}{x}) \le x^4$

and $\lim_{x \rightarrow 0}-x^4=\lim_{x \rightarrow 0}x^4=L$, then we can also conclude that $\im_{x \rightarrow} x^4\cos(\frac{2}{x})=L$.

So we can actually evaluate the "if" limits pretty easily since both are continuous  and exist at $x=0$.

$\lim_{x \rightarrow 0}x^4=0^4=0$

$\lim_{x \rightarrow 0}-x^4=-0^4=-0=0$.

We can finally conclude that $\lim_{\rightarrow 0}x^4\cos(\frac{2}{x})=0$ by squeeze theorem.

Some people call this sandwich theorem.