Example. Compute the limit

$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}
$$
Solution.

1. As $n \rightarrow \infty$, the expression under the limit sign represents a power, the base of which tends to one:

$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)=1
$$

and the exponent tends to minus infinity:

$$
\lim _{n \rightarrow \infty}(1-2 n)=-\infty
$$

We transform the expression under the limit sign so that we can use the second remarkable limit:

$$
\begin{aligned}
& \left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}=\left(1+\frac{2 n-4}{4 n^{2}+2 n+3}\right)^{1-2 n}= \\
& =\left[\left(1+\frac{2 n-4}{4 n^{2}+2 n+3}\right)^{\frac{4 n^{2}+2 n+3}{2 n-4}}\right]^{\frac{(2 n-4)(1-2 n)}{4 n^{2}+2 n+3}}
\end{aligned}
$$

Since

$$
\frac{2 n-4}{4 n^{2}+2 n+3} \rightarrow 0
$$

as $n \rightarrow \infty$, then

$$
\lim _{n \rightarrow \infty}\left(1+\frac{2 n-4}{4 n^{2}+2 n+3}\right)^{\frac{4 n^{2}+2 n+3}{2 n-4}}=e
$$

2. Since

$$
\lim _{n \rightarrow \infty} \frac{(2 n-4)(1-2 n)}{4 n^{2}+2 n+3}=-1
$$

we finally have

$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}=e^{-1}
$$

Answer. $\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}=e^{-1}$.

Conditions of the Problems. Calculate the limits.

1. $\lim _{n \rightarrow \infty}\left(\frac{n+2}{n-1}\right)^{n}$.
2. $\lim _{n \rightarrow \infty}\left(\frac{2 n+4}{2 n+3}\right)^{n+1}$.
3. $\lim _{n \rightarrow \infty}\left(\frac{n^{2}+1}{n^{2}}\right)^{n^{2}}$.
4. $\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+3}{2 n^{2}+1}\right)^{n^{2}}$.
5. $\lim _{n \rightarrow \infty}\left(\frac{n^{2}+n+3}{n^{2}+n-1}\right)^{-n^{2}}$.
6. $\lim _{n \rightarrow \infty}\left(\frac{n+7}{n+5}\right)^{n+3}$.
7. $\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-2 n}{3 n^{2}-2 n+5}\right)^{n+2}$.
8. $\lim _{n \rightarrow \infty}\left(\frac{2 n^{2}+n+5}{2 n^{2}+n+1}\right)^{3 n^{2}}$.
9. $\lim _{n \rightarrow \infty}\left(\frac{n^{3}+3}{n^{3}-2}\right)^{n-n^{3}}$.
10. $\lim _{n \rightarrow \infty}\left(\frac{n+1}{n-1}\right)^{3 n^{2}+1}$.

Answers. 1. $e^{3}$. 2. $\sqrt{e}$. 3. e. 4. e. 5. $e^{-2}$. 6. $e^{2}$. 7. 1. 8. $e^{6}$. 9. $e^{-5}$. 10. $+\infty$.

## 3.5. The Concept of the Limit of a Function

Problem Statement. Using the definition of the limit of a function at a point, prove that

$$
\lim _{x \rightarrow a} f(x)=A
$$

Plan of Solution.

1. A number $A$ is called the limit of the function $f(x)$ at the point $x=a$ if

$$
\forall \varepsilon>0 \quad \exists \delta(\varepsilon)>0: \quad 00$ the inequality $|f(x)-A|0 \quad 0<|x-a|<\delta(\varepsilon) \Longrightarrow|f(x)-A|<\varepsilon$.