Limit of piecewise function and find a

Question

I calculated the limit for 1 ^ - and found it infinite. But for 1 ^ + I can't find a when the 2nd function equals infinity. What solution should I do?

Answer 1

Right limit at $x=1$ is $$RL=\lim_{h\to 0} [2(1+h)-4+3^\frac{1+h}{1-(1+h)^2}$$ $$RL=-2+\lim_{h \to 0}3^{\frac{(1+h)}{1-(1+2h)}}=-2+\lim_{h\to 0}3^{\frac{1+h}{-2h}}=-2$$ Left limit at $x=1$ is $$LL=\frac{8}{\pi}\tan^{-1}a $$ Finally, $LL=RL$ gives $a=-1$.