I am given the piecewise function $G(t)$, which is written below, and I am asked to find all values of the parameters $\alpha$ and $\beta$ for which the function $G$ is differentiable at $t=1$.
$G(t)=\alpha t^{2}+\ln(t)$ if $t\geq 1$
$G(t)=\beta e^{t-1}-2t$ if $t<1$
I am also given the hint:
You may assume that:$\lim_{x\to0} \frac{e^{x}-1}{x}=1$ $\lim_{x\to 0} \frac{\ln(1+x)}{x}=1$
I know that the function must be continuous, so $\lim\limits_{t\to 1^+}G(t)=\lim\limits_{t\to 1^-}G(t)$. This leaves me with $\beta-2=\alpha$. I am a bit confused with my next steps. I believe I must use the definition of a derivative in order to set the slopes of each parts of $G(t)$ to be equal to each other. We are not far enough in the course to simply take the derivatives of each part of $G(t)$.