If the function
Question:

If the function

$f(x)=\left\{\begin{array}{l}a|\pi-x|+1, x \leq 5 \\ b|x-\pi|+3, x>5\end{array}\right.$

is continuous at $x=5$, then the value of $a-b$ is:

1. (1) $\frac{2}{\pi+5}$

2. (2) $\frac{-2}{\pi+5}$

3. (3) $\frac{2}{\pi-5}$

4. (4) $\frac{2}{5-\pi}$

Correct Option: , 4

Solution:

R.H.L. $\lim _{x \rightarrow 5^{+}} b|(x-\pi)|+3=(5-\pi) b+3$

$f(5)=$ L.H.L. $\lim _{x \rightarrow 5^{-}} a|(\pi-x)|+1=a(5-\pi)+1$

$\because$ function is continuous at $x=5$

$\therefore \mathrm{LHL}=\mathrm{RHL}$

$(5-\pi) b+3=(5-\pi) a+1$

$\Rightarrow 2=(a-b)(5-\pi) \Rightarrow a-b=\frac{2}{5-\pi}$