Question

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

Answer 1

Answer: 8

$\underset{x\to 2}{\lim }\left(\frac{2^x+2^{3-x}-6}{\sqrt{2^{-x}}+2^{1-x}}\right)$
$=\underset{x\to 2}{\lim }\frac{2^{2x}+2^3-6\left(2^x\right)}{\sqrt{2^x}-2}$
$=\underset{x\to 2}{\lim }\frac{\left(2^x-2\right)\left(2^x-4\right)\left(\sqrt{2^x}+2\right)}{2^x-4}$
$=\underset{x\to 2}{\lim }\left[\left(2^x-2\right)\left(\sqrt{2^x}+2\right)\right]$
$=8$