Limit of a function at a point, undefined?

Question

Limit of $\cos\left(\frac{1}{(1-x)^2}\right)$ when $x=1$. I believe this function might be undefined at this point due to the $\frac{1}{0}$ which appears, however I'm not sure if this is correct and if so how to show that.

Answer 1

When we take the limit we are assuming that $x\neq 1$ then it doesn't matter if the function is not defined for $x=1$.

Take also a look here Why are we allowed to cancel fractions in limits?.

For the limit $\cos\left( \frac{1}{(1-x)^2} \right)$the problem is that $\frac{1}{(1-x)^2} \to \infty$ and $\lim_{t\to \infty} \cos t$ doesn't exist.

To show that let consider

• $t_n=2\pi n\to \infty \implies \cos (t_n)\to 1$

but

• $t_n=(2n+1) \pi\to \infty \implies \cos (t_n)\to -1$