How do I compute this limit?

Question

For the following function $f$ and point $a$, determine whether $\lim \limits_{x \to a} f (x)$ exists, and compute the limit if it exists. Justify your answer.

$f(x)=\cos(\frac{1}{(1-x)^2})$ where $a=1$

This is a 3 mark question on a past paper so it shouldn't be too complex. I suspect you have to manipulate the $\frac{1}{(1-x)^2}$ somehow, but I just can't see it.

Answer 1

Let consider as $n\to \infty$

• $x_n=1-\sqrt{\frac1{n\pi}}\to 1 \implies f(x_n)=\cos(n\pi)=\begin{cases} 1\quad \text{for n even}\\-1\quad \text{for n odd}\end{cases}$

then there exist two subsequences with different limit and therefore the limit doesn't exist.

Answer 2

Hint:$$\begin{aligned}\lim_{x\to 1}\cos\left(\frac1{(1-x)^2}\right)&=\lim_{x\to 0}\cos\left(\frac1{x^2}\right)\\ &=\lim_{x\to \pm\infty}\cos\left(x^2\right)\end{aligned}$$