Example 14. Find $\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{2 \operatorname{ctg}^{2} x}$.
Solution. Since $\operatorname{ctg} x=\frac{1}{\operatorname{tg} x}$, then

$$
\begin{gathered}
\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{2 \operatorname{ctg}^{2} x}=\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{\frac{2}{\operatorname{tg}^{2} x}}= \\
=\lim _{x \rightarrow 0}\left[\left(1+\operatorname{tg}^{2} x\right)^{\frac{1}{\mathrm{gg}^{2} x}}\right]^{2}=\left[\lim _{x \rightarrow 0}\left(1+\operatorname{tg}^{2} x\right)^{\frac{1}{\lg ^{2} x}}\right]^{2}=e^{2}
\end{gathered}
$$

## Problems

Find the limits:

1. $\lim _{n \rightarrow \infty}\left(1+\frac{2}{n}\right)^{n}$.
2. $\lim _{x \rightarrow 0}(1-3 x)^{\frac{1}{x}}$.
3. $\lim _{x \rightarrow 0} \frac{\ln (1+4 x)}{x}$.
4. $\lim _{x \rightarrow \infty} x[\ln (x+1)-\ln x]$.
5. $\lim _{x \rightarrow \infty}\left(\frac{x+1}{x-3}\right)^{x}$.
6. $\lim _{x \rightarrow \infty}\left(\frac{x+n}{x+m}\right)^{x}$.
7. $\lim _{x \rightarrow \infty} \frac{\sin \frac{x}{5}}{x}$.
8. $\lim _{x \rightarrow 0} x \sin \frac{1}{x}$.
9. $\lim _{x \rightarrow 0} \frac{\operatorname{arctg} x}{x}$.
10. $\lim _{x \rightarrow 0} \frac{\sin a x}{\operatorname{tg} b x}$.
11. $\lim _{x \rightarrow 0} \frac{\sin ^{3} \frac{x}{2}}{x^{3}}$.
12. $\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+4}-2}$.
13. $\lim _{x \rightarrow 1} \frac{\sin (x-1)}{x^{3}-1}$.
14. $\lim _{x \rightarrow 0} \frac{1-\cos x-\operatorname{tg}^{2} x}{x \sin x}$.
15. $\lim _{x \rightarrow 0}\left(\frac{\sin \frac{x}{2}}{x}\right)^{x+3}$.
16. $\lim _{x \rightarrow \infty}\left(\frac{2 x+1}{4 x-3}\right)^{x}$.
17. $\lim _{x \rightarrow 0}(\cos x)^{\frac{1}{x^{2}}}$.
18. $\lim _{x \rightarrow \frac{\pi}{2}}(\sin x)^{\operatorname{tg}^{2} x}$.

## Answers

1. $e^{2}$. 2. $e^{-3}$. 3. 4. Hint. Let $4 x=\alpha$ and use formula (1.65). 4.1. Hint. Let $\frac{1}{x}=\alpha$. Use formula (1.65). 5. $e^{4}$. 6. $e^{n-m}$. 7. $\frac{1}{5}$. 8. 1. Hint. Let $\frac{1}{x}=\alpha$. 9. 1. Hint. Let $\operatorname{arctg} x=\alpha$. 10. $\frac{a}{b}$. 11. $\frac{1}{8}$. 12. 4. 13. $\frac{1}{3}$. 14. $-\frac{1}{2}$. 15. $\frac{1}{8}$. 16. 0. 17. $e^{-\frac{1}{2}}$. 18. $e^{-\frac{1}{2}}$. Hint. Let $\cos ^{2} x=\alpha$.

## § 1.6. Various Examples of Finding Limits

When finding limits, indeterminate forms of the type $(\infty-\infty)$ and $0 \cdot \infty$ may be encountered. These cases are transformed into one of the two considered cases, i.e., indeterminate forms of the type $\frac{0}{0}$ or $\frac{\infty}{\infty}$. We will show on examples how such limits are found.