Consider the following statements in respect of a function f(x):

1. f(x) is continuous at x = a, if lim_x→a f(x) exists.

2. If f(x) is continuous at a point, then 1/f(x) is also continuous at that point.

Which of the above statements is/are correct?

Concept:

\( \displaystyle\underset{x\to a}{\mathop{\lim }}\rm\,f\left( x \right)\) exists if \(\rm\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)\)

f(x) is Continuous at x = a ⇔\(\rm\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=lim _{x \rightarrow a} f(x)\)

Calculation:

1. This statement is false, as the limit at the given point should be equal to the existence of the limit. 

2. If f(x) is a continuous function at a point, then it is not necessary that the function 1/ f(x)  will be continuous.

Take, f (x) = x, which is continuous at a point, 

For 1/f (x) = 1/x, which will not be continuous at the same point x = 0

So, this statement is not true. 

Hence, option (4) is correct.