Example 2. Find $\lim _{x \rightarrow 1} 3^{(2+x) /(x-1)}$.
Solution.

$\lim _{x \rightarrow+1-0} 3^{(2+x) /(x-1)}=\left\{\begin{array}{l}\text { the exponent in absolute value increases, } \\ \text { but its sign is negative; the value under } \\ \text { the limit sign decreases, remaining } \\ \text { positive }\end{array}\right\}=0 ;$
$\lim _{x \rightarrow 1+0} 3^{(2+x) /(x-1)}=\left\{\begin{array}{l}\text { the exponent is positive and increases without } \\ \text { bound; the value under the limit sign increases } \\ \text { without bound, remaining positive }\end{array}\right\}=+\infty$.

Therefore, $\lim _{x \rightarrow 1} 3^{(2+x) /(x-1)}$ does not exist.