Solution
= Solution
Recognize the <derivative> of the <exponential function> at $x=1$:
$$
\boxed{\lim_{x\to1}\frac{2^x-2}{x-1}=\left.\frac{d}{dx}2^x\right|_{x=1}=2\ln2.}
$$
Here $2^x=e^{x\ln2}$ and the <chain rule> supplies the factor $\ln2$.
= Solution
Recognize the <derivative> of the <exponential function> at $x=1$:
$$
\boxed{\lim_{x\to1}\frac{2^x-2}{x-1}=\left.\frac{d}{dx}2^x\right|_{x=1}=2\ln2.}
$$
Here $2^x=e^{x\ln2}$ and the <chain rule> supplies the factor $\ln2$.