Example. Compute the limit

$$
\lim _{x \rightarrow 0} \sqrt[3]{x\left(2+\sin \frac{1}{x}\right)+8 \cos x}
$$
Solution.

1. Since the function $y=\sqrt[3]{x}$ is continuous for all $x$, by passing to the limit under the sign of a continuous function, we get

$$
\lim _{x \rightarrow 0} \sqrt[3]{x\left(2+\sin \frac{1}{x}\right)+8 \cos x}=\sqrt[3]{\lim _{x \rightarrow 0}\left[x\left(2+\sin \frac{1}{x}\right)+8 \cos x\right]}
$$

2. Since $x$ is an infinitesimal function at the point $x=0$, and $2+\sin (1 / x)$ is a bounded function in the neighborhood of the point $x=0$, then $x(2+\sin (1 / x))$ is an infinitesimal function at the point $x=0$, i.e.,

$$
\lim _{x \rightarrow 0} x\left(2+\sin \frac{1}{x}\right)=0
$$

3. Since $\cos x$ is continuous at the point $x=0$, then

$$
\lim _{x \rightarrow 0} \cos x=1
$$

and, using the properties of the limit of a function at a point, we get

$$
\begin{aligned}
& \lim _{x \rightarrow 0} \sqrt[3]{x\left(2+\sin \frac{1}{x}\right)+8 \cos x}=\sqrt[3]{\lim _{x \rightarrow 0} x\left(2+\sin \frac{1}{x}\right)+8 \lim _{x \rightarrow 0} \cos x}=2 \\
& \text { Answer. } \lim _{x \rightarrow 0} \sqrt[3]{x\left(2+\sin \left(\frac{1}{x}\right)\right)+8 \cos x}=2
\end{aligned}
$$

Conditions of the Problems. Calculate the limit.

1. $\lim _{x \rightarrow 0} \sqrt{9 \cos 2 x+2 x \operatorname{arctg} \frac{1}{x}}$.
2. $\lim _{x \rightarrow \pi / 2} \sqrt{4 \sin x+(2 x-\pi) \sin \frac{x^{2}}{2 x-\pi}}$.
3. $\lim _{x \rightarrow 0} \sqrt{\cos x+\operatorname{arctg} x \cos ^{2} \frac{1}{x^{2}}}$.
4. $\lim _{x \rightarrow 0} \sqrt{4 \cos x+\ln (1+2 x) \sin \frac{1}{x}}$.
5. $\lim _{x \rightarrow 0} \sqrt{4 \cos ^{2} x+\left(e^{2 x}-1\right) \operatorname{arctg} \frac{1}{x^{2}}}$.
6. $\lim _{x \rightarrow 0} \ln \left[\left(e^{x^{3}}-\cos x\right) \cos \frac{2}{x}+\operatorname{tg}\left(x+\frac{\pi}{4}\right)\right]$.
7. $\lim _{x \rightarrow 0} \sqrt{4 \sin x+\left(e^{\sin ^{2} x}-1\right) \cos \frac{1}{x}}$.
8. $\lim _{x \rightarrow 2} \sqrt{\ln (x+2)+\sin \left(4-x^{2}\right) \cos \frac{x+2}{x-2}}$.
9. $\lim _{x \rightarrow 1} \operatorname{tg}\left(\arccos x+\sin \frac{x-1}{x+1} \cos \frac{x+1}{x-1}\right)$.
10. $\lim _{x \rightarrow 0} \ln \left(3+\operatorname{arctg} x \sin \frac{1}{x}\right)$.
Answers. 1. 3.
11. 2 .
12. 1 .
13. 2 .
14. 2 .
15. $0 . \quad 7.2$.
16. $\ln 2$.
17. $0 . \quad 10 . \ln 3$.

Chapter 4

## DIFFERENTIATION

In the study of the topic DIFFERENTIATION, you will become familiar with the concepts of the derivative and differential of a function of one variable through examples, learn to compute derivatives using the rules of differentiation for sums, products, quotients, and composite functions, learn to differentiate functions given parametrically, compute higher-order derivatives, and apply derivatives and differentials in approximate calculations and in solving geometric problems.

Using the package SOLVER.CA, you can compute limits, perform numerical calculations, and compute derivatives of any order to verify the correctness of the results you obtain.

## 4.1. The Concept of a Derivative

Problem Statement. Find the derivative of the function $f(x)$ at the point $x=0$ based on the definition.

Plan of Solution.

1. By definition,

$$
f^{\prime}(0)=\lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x}
$$

(Recall that when computing the limit $x \rightarrow 0$, but $x \neq 0$.)

2. Compute the limit

$$
\lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x}
$$

3. If the limit exists and equals $A$, then $f^{\prime}(0)=A$; if the limit does not exist, then $f^{\prime}(0)$ does not exist.