Answer
$$\lim _{x \rightarrow 0}\left[e^{x}-\ln (x+1)\right] =1$$
Work Step by Step
Given $$\lim _{x \rightarrow 0}\left[e^{x}-\ln (x+1)\right]$$
By using the method of replacement
\begin{align*}
\lim _{x \rightarrow 0}\left[e^{x}-\ln (x+1)\right]&=\lim _{x \rightarrow 0}\left[e^{0}-\ln (0+1)\right]\\
&= \lim _{x \rightarrow 0}\left[1 \right]\\
&=1
\end{align*}