Answer:
Proof in explanation.
Step-by-step explanation:
I'm going to attempt this by squeeze theorem.
We know that [tex]\cos(\frac{2}{x})[/tex] is a variable number between -1 and 1 (inclusive).
This means that [tex]-1 \le \cos(\frac{2}{x}) \le 1[/tex].
[tex]x^4 \ge 0[/tex] for all value [tex]x[/tex]. So if we multiply all sides of our inequality by this, it will not effect the direction of the inequalities.
[tex]-x^4 \le x^4 \cos(\frac{2}{x}) \le x^4[/tex]
By squeeze theorem, if [tex]-x^4 \le x^4 \cos(\frac{2}{x}) \le x^4[/tex]
and [tex]\lim_{x \rightarrow 0}-x^4=\lim_{x \rightarrow 0}x^4=L[/tex], then we can also conclude that [tex]\im_{x \rightarrow} x^4\cos(\frac{2}{x})=L[/tex].
So we can actually evaluate the "if" limits pretty easily since both are continuous and exist at [tex]x=0[/tex].
[tex]\lim_{x \rightarrow 0}x^4=0^4=0[/tex]
[tex]\lim_{x \rightarrow 0}-x^4=-0^4=-0=0[/tex].
We can finally conclude that [tex]\lim_{\rightarrow 0}x^4\cos(\frac{2}{x})=0[/tex] by squeeze theorem.
Some people call this sandwich theorem.
