Question

Solve: $\underset{x\to 2}{lim}\frac{(1-3^x-4^x+{12}^x)}{\sqrt{(2\cos x+7)}-}$

Answer 1

$\underset{x\to 2}{lim}\frac{(1-3^x-4^x+{12}^x)}{\sqrt{2\cos x+7}}$

$=\left(\underset{x\to 2}{lim}1-\underset{x\to 2}{lim}{3}^x-\underset{x\to 2}{lim}{4}^x+\underset{x\to 2}{lim}{12}^x\right)\times \frac{1}{\sqrt{2\underset{x\to 2}{lim}\cos x-\underset{x\to 2}{lim}7}}$

$=\frac{1-3^2-4^2+{12}^2}{\sqrt{2\cos 2-7}}$

$=\frac{1-9-16+144}{\sqrt{2\cos 2-7}}$

$=\frac{120}{\sqrt{2\cos 2-7}}$

$\therefore \underset{x\to 2}{lim}\frac{1-3^x-4^x+{12}^x}{\sqrt{2\cos x+7}}=\frac{120}{\sqrt{2\cos 2-7}}$.