Example 2.22. Find the limit

$$
\lim _{x \rightarrow 0}\left(\int_{0}^{\operatorname{arctg} x} e^{\sin x} d x / \int_{0}^{x} \cos \left(x^{2}\right) d x\right)
$$
Solution. According to L'Hôpital's rule and the rules for differentiating definite integrals, we transform the limit as follows:

$$
\lim _{x \rightarrow 0}\left(\int_{0}^{\operatorname{arctg} x} e^{\sin x} d x\right)^{\prime} /\left(\int_{0}^{x} \cos \left(x^{2}\right) d x\right)^{\prime}=\lim _{x \rightarrow 0} \frac{e^{\sin (\operatorname{arctg} x)}}{\left(1+x^{2}\right) \cos x^{2}}=1
$$

## 2.3 .  IMPROPER INTEGRALS