I'm having trouble identifying some limits, or actually, I figured out what they are just not the argument for why. Here they go:
$\lim_{a\to1}\frac{1}{a}\left(x\cdot\left(1-a\right)^{\frac{1-a}{a}}+b\left(1-a\right)^{\frac{1}{a}}\right)^{a}=x $
And the limit: $\lim_{a\to-\infty}\frac{1-a}{a}\left(\frac{bx}{1-a}+1\right)^{a}=-e^{-bx}$
So the first is easy if one may evaluate the terms in parts. I know that if $a_{n},b_{n}$ are converges to a,b then $a_{n}b_{n}\to ab\quad a_{n}+b_{n}\to a+b$ but i'm not sure how to handle the power. The second one I feel like doing the good old $e^{\ln(x)}=x$ trick which gives: $e^{a\cdot\left(\left(\ln\left(\frac{bx}{1-a}\right)+1\right)+\ln\left(\frac{1-a}{a}\right)\right)}$
But it doesn't really work out, the second terms looks delicious since it's limit on its self is $\pi \iota$ but the "a" in front kind of messes that up and the first part looks like something that could easily be l'hospital'ed but I just can't get the last steps.
Hope someone is willing to help.