Question

The ordered pair (a,b) such that $f\left(x\right)=\{\begin{array}{ccc}\frac{b{e}^x-cosx-x}{x}& ,& x>0\\ a& ,& x=0\\ \frac{2ta{n}^{-1}\left(e^x\right)-\frac{\pi }{4}}{x}& ,& x<0\end{array}$ becomes continuous at x-0 is

Answer 1

$f\left(x\right)=\{\begin{array}{ccc}\frac{b{e}^x-cosx-x}{x}& ,& x>0\\ a& ,& x=0\\ \frac{2ta{n}^{-1}\left(e^x\right)-\frac{\pi }{4}}{x}& ,& x<0\end{array}$

for $x>0,{lim}_{x\to 0}\frac{b{e}^x-cosx-x}{x}$

for this limit it be finite, $b-1=0,b=1$

for $f\left(x\right)$ to be continuous, ${lim}_{x\to 0}\frac{b{e}^x-cosx-x}{x}=a$

applying L'hospital rule, ${lim}_{x\to 0}\frac{b{e}^x-cosx-x}{x}={lim}_{x\to 0}b{e}^x+\sin x=b=a=1$