136. Find $\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}-2 x+3}}$.
Solution. The limit transition as $x \rightarrow \infty$ can always be replaced by the limit transition as $\alpha \rightarrow 0$, if we set $\alpha=1 / x$ (the method of variable substitution).

Thus, setting $x=1 / \alpha$ in this case, we find that $\alpha \rightarrow 0$ as $x \rightarrow \infty$. Therefore,

$$
\begin{aligned}
\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}-2 x+3}}= & \lim _{\alpha \rightarrow 0} \frac{3 \cdot \frac{1}{\alpha}}{\sqrt{\frac{1}{\alpha^{2}}-2 \cdot \frac{1}{\alpha}+3}}=\lim _{\alpha \rightarrow 0} \frac{3 \cdot \frac{1}{\alpha}}{\sqrt{\frac{1-2 \alpha+3 \alpha^{2}}{\alpha^{2}}}}= \\
& =\lim _{\alpha \rightarrow 0} \frac{3}{\sqrt{1-2 \alpha+3 \alpha^{2}}}=\frac{3}{1}=3
\end{aligned}
$$