Question

Solve
${lim}_{x\to 2}\left(\frac{3^x+3^{3-x}-12}{3^{3-x}-3^{x/2}}\right)$

Answer 1

$\begin{array}{c}{lim}_{x\to 2}\left(\frac{3^x+3^{3x}-12}{3^{3x}-3^{\frac{x}{2}}}\right)\\ Itis\frac{0}{0}form\\ \Rightarrow {lim}_{x\to 2}\frac{3^x\log 3+{27.3}^{-x}{\log }_3}{-27\times 3^{-x}{\log }_3-3^{\frac{x}{2}}\times \frac{1}{2}{\log }_3}\\ \Rightarrow {lim}_{x\to 2}\frac{3^x+{27.3}^{-x}}{-{27.3}^{-x}\frac{-1\times 3^{\frac{x}{2}}}{2}}\\ \Rightarrow {lim}_{x\to 2}\frac{9+\frac{27}{9}}{-\left(\frac{27}{9}+\frac{3}{2}\right)}\frac{-12}{\left(3+\frac{3}{2}\right)}=\frac{-24}{9}=\frac{-8}{3}Ans.\end{array}$