$\lim _{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}=$

Question

Aicharya · Accepted Answer

$\begin{aligned} & \lim _{x ightarrow 2} \frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}} \ & =\lim _{x ightarrow 2} \frac{\left(3^xight)^2+3^3-12\left(3^xight)}{3^3-\left(3^{\frac{x}{2}}ight)^3} \ & =\lim _{x ightarrow 2} \frac{\left(3^x-9ight)\left(3^x-3ight)}{\left(3-3^{\frac{x}{2}}ight)\left(9+3.3^{\frac{x}{2}}+3^xight)} \ & =-\lim _{x ightarrow 2} \frac{\left(3^{\frac{x}{2}}+3ight)\left(3^{\frac{x}{2}}-3ight)\left(3^x-3ight)}{\left(3^{\frac{x}{2}}-3ight)\left(3^x+3.3^{\frac{x}{2}}+9ight)} \ & =-\frac{(3+3)(9-3)}{9+9+9}\end{aligned}$ $=\frac{-4}{3}$

$\lim _{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}=$

Solution