Question

Question 5: prove that it’s =0

Answer 1

Proof in explanation.

Explanation:

I'm going to attempt this by squeeze theorem.

We know that
`\cos((2)/(x))` is a variable number between -1 and 1 (inclusive).

This means that
`-1 \le \cos((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((2)/(x)) \le x^4`

By squeeze theorem, if
`-x^4 \le x^4 \cos((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((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((2)/(x))=0` by squeeze theorem.

Some people call this sandwich theorem.