functions limits

Question

In question No.9 why we can not conclude anything about f at x=1 ? I did not really get it so can someone explain it for me ?

Answer 1

In the definition we have $$\forall \epsilon>0,\exists \delta, 0<|x-x_0|<\delta\to |f(x)-L|<\epsilon$$for some $L$ where nowhere restricts the value of the function at the exact point. For example consider the following function$$f(x)=\begin{cases}\dfrac{5x-5}{x-1}&x\ne 1\\a&x=1\end{cases}$$for different values of $a$. In fact if any function satisfies $$\lim_{x\to a}f(x)=f(a)$$the function is referred to as $\text{continuous at }x=a$

Answer 2

The definition of limit at $x_0$ is independent of the value of the function at $x_0$. Check the definition!

Answer 3

Consider the function

$$f(x)~=~\begin{cases} 5x~\text{, for } x\in \mathbb{R}/\{1\} \\ 42~\text{, for } x=1 \\ \end{cases} $$

while the limit for sure goes for $x=1$ to $5$ the functions is defined in another way at this point. You can construct functions like this with ease and there is the problem.