156. Find $\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$.
Solution. We have

$$
\begin{aligned}
& \lim _{x \rightarrow e} \frac{\ln x-1}{x-e}=\lim _{x \rightarrow e} \frac{\ln x-\ln e}{x-e}=\frac{1}{e} \lim _{x \rightarrow e} \frac{\ln \frac{x}{e}}{\frac{x}{e}-1}= \\
= & \frac{1}{e} \lim _{z \rightarrow 0} \frac{\ln (z+1)}{z}=\frac{1}{e} \cdot 1=\frac{1}{e}\left(\text { here } \frac{x}{e}-1=z\right) .
\end{aligned}
$$

$157-160$. Find the limits:

$\begin{array}{ll}\text { 157. } \lim _{x \rightarrow \infty}\left(1+\frac{5}{x}\right)^{x} & \text { 158. } \lim _{x \rightarrow \infty}\left(\frac{x+2}{x}\right)^{2 x} . \\ \text { 159. } \lim _{x \rightarrow \infty} \frac{\ln (1+x)}{x} . & \text { 160. } \lim _{x \rightarrow \infty}\left(1+\frac{2}{3 x}\right)^{3 x} .\end{array}$

## § 3. Derivative

## Problems Leading to the Concept of Derivative  Definition of Derivative  General Rule for Finding the Derivative  Particular Value of the Derivative  Connection Between Continuity and Differentiability of a Function