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:

  1. 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.



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