## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 10} \frac{\lg x-1}{\sqrt{x-9}-1}$
## Solution

Substitution:

$$
\begin{aligned}
& x=10(y+1) \Rightarrow y=\frac{x}{10}-1 \\
& x \rightarrow 10 \Rightarrow y \rightarrow 0
\end{aligned}
$$

We obtain:

$$
\begin{aligned}
& \lim _{x \rightarrow 10} \frac{\lg x-1}{\sqrt{x-9}-1}=\lim _{y \rightarrow 0} \frac{\lg 10(y+1)-1}{\sqrt{10(y+1)-9}-1}= \\
& =\lim _{y \rightarrow 0} \frac{\lg 10+\lg (y+1)-1}{\sqrt{10 y+10-9}-1}= \\
& =\lim _{y \rightarrow 0} \frac{1+\lg (y+1)-1}{\sqrt{10 y+1}-1}=\lim _{y \rightarrow 0} \frac{\lg (y+1)}{\sqrt{10 y+1}-1}= \\
& =\lim _{y \rightarrow 0} \frac{\ln (y+1)}{\ln 10(\sqrt{10 y+1}-1)}=
\end{aligned}
$$

Using the substitution of equivalent infinitesimals:

$\ln (1+y) \sim y_{\text {, as }} y \rightarrow 0$

We get:

$=\lim _{y \rightarrow 0} \frac{y}{\ln 10(\sqrt{10 y+1}-1)}=\lim _{y \rightarrow 0} \frac{y(\sqrt{10 y+1}+1)}{\ln 10(\sqrt{10 y+1}-1)(\sqrt{10 y+1}+1)}=$

$$
\begin{aligned}
& =\lim _{y \rightarrow 0} \frac{y(\sqrt{10 y+1}+1)}{\ln 10(10 y+1-1)}= \\
& =\lim _{y \rightarrow 0} \frac{y(\sqrt{10 y+1}+1)}{\ln 10 \cdot 10 y}=\lim _{y \rightarrow 0} \frac{\sqrt{10 y+1}+1}{10 \ln 10}= \\
& =\frac{\sqrt{10 \cdot 0+1}+1}{10 \ln 10}=\frac{2}{10 \ln 10}=\frac{1}{5 \ln 10}
\end{aligned}
$$

## Problem Kuznetsov Limits 16-25