Which of the following function need not necessarily has limit? If $f(x)$ is a function such that $\lim_{x\to a}f(x)=L$ exists,then which of the following function need not necessarily has limit?
$1.|f(x)-f(a)|$
$2.\frac{\sin(f(x))}{f(x)}$
$3.log(f(x))$
$4.e^{-f(x)}$
My attempt:I think $3$ option is the answer if it is base ten logarithmic,but for natural log this exists.What should be my answer and why?
Thanks.
 A: Presumably these are all limits as $x \to a$?
If so, the main result you want to use is "if $g(y)$ is continuous at $L$ and $\lim_{x\to a}f(x) = L$, then $\lim_{x\to a}g(f(x)) = g(L) = g\left(\lim_{x\to a} f(x)\right)$".
Given this, which of the above functions might fail to be continuous at $L$? (note this will depend on what $L$ is, so you should state the different cases for each). Further, you should worry a little about $f(a)$ in the first expression.
More detailed hints/answers:


*

*$g(y) = |y - f(a)|$ is continuous for all $y$, provided $f(a)$ is defined (the limit exists, but the function may still be undefined).


*$g(y) = \frac{\sin(y)}{y}$ is continuous, provided $y \neq 0$. So this limit exists if $L \neq 0$, otherwise it is possible it doesn't. However, the limit of $\sin(x)/x$ as $x \to 0$ is well-known and converges to $1$.


*$g(y) = \log(y)$ is continuous for $y > 0$. So this limit exists if $L > 0$, otherwise it doesn't.


*$e^{-y}$ is continuous for all $y$. So this limit always exists.
Thus, 1 and 3 could all fail to have limits. 2 and 4 are the ones that are guaranteed.

Actually, more precisely, 1. could fail to be well-defined, if $f(a)$ is not defined. This is distinct from not having a limit. I wonder if the exercise wants you to explore this or not.
A: Just a tip: if the $f(x) \to 0$ what will happen with the $\log(f(x))$? 
