Limit Continuity And Differentiability Ques 101

Answer:

Correct Answer: 101.(d)

Solution:

101. As, $f(x)$ is continuous and $g(x)$ is discontinuous.

Case I $g(x)$ is discontinuous as limit does not exist at $x=k$.

$\therefore \quad \varphi(x)=f(x)+g(x)$

$\Rightarrow \lim _{x \rightarrow k} \varphi(x)=\lim _{x \rightarrow k}{f(x)+g(x)}=$ does not exist.

$\therefore \varphi(x)$ is discontinuous.

Case II $g(x)$ is discontinuous as, $\lim _{x \rightarrow k} g(x) \neq g(k)$.

$\therefore$

$$ \varphi(x)=f(x)+g(x) $$

$\Rightarrow \lim _{x \rightarrow k} \varphi(x)=\lim _{x \rightarrow k}{f(x)+g(x)}=$ exists and is a finite quantity

but $\varphi(k)=f(k)+g(k) \neq \lim _{x \rightarrow k}{f(x)+g(x)}$

$\therefore \quad \varphi(x)=f(x)+g(x)$ is continuous,

whenever $g(x)$ is discontinuous.