Question:
$\lim _{x \rightarrow 2} \frac{3^{x}+3^{3-x}-12}{3^{-x / 2}-3^{1-x}}$ is equal to
Solution:
Let $3^{x}=t^{2}$
$\lim _{t \rightarrow 3} \frac{t^{2}+\frac{27}{t^{2}}-12}{\frac{1}{t}-\frac{3}{t^{2}}}$
$=\lim _{t \rightarrow 3} \frac{t^{4}-12 t^{2}+27}{t-3}$
$=\lim _{t \rightarrow 3} \frac{\left(t^{2}-3\right)(t+3)(t-3)}{t-3}$
$=\left(3^{2}-3\right)(3+3)=36$