Limit Continuity and Differentiability 5 Question 3
3. If the function $f(x)=\begin{aligned} & a|\pi-x|+1, x \leq 5 \ & b|x-\pi|+3, x>5\end{aligned}$ is continuous at $x=5$, then the value of $a-b$ is
(a) $\frac{-2}{\pi+5}$
(b) $\frac{2}{\pi+5}$
(c) $\frac{2}{\pi-5}$
(d) $\frac{2}{5-\pi}$
(2019 Main, 9 April II)
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Answer:
Correct Answer: 3. (d)
Solution:
- 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 discontinuous,
whenever $g(x)$ is discontinuous.
