Question

The value of $\underset{x\to 2}{lim}\frac{3^{x/2}-3}{3^x-9}$ is

• A. $0$
• B. $1/3$
• C. $1/6$
• D. $\ln 3$

Answer 1

Answer: (C)

$1/6$

$\underset{x\to 2}{lim}\left(\frac{3^{x/2}-3}{3^x-9}\right)$
$=\underset{x\to 2}{lim}\left(\frac{3^{x/2}-3}{{\left(3^{x/2}\right)}^2-3^2}\right)$
$=\underset{x\to 2}{lim}\frac{3^{x/2}-3}{(3^{x/2}-3)(3^{x/2}+3)}$
$=\underset{x\to 2}{lim}\frac{1}{3^{x/2}+3}$
$=\frac{1}{3^1+3}=\frac{1}{6}$