Solveeit Logo
Login

Question

Question: If \(f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x};(x \neq 0),\) is continuous function at \(x = 0\), then...

If f(x)=2−x+4sin⁡2x;(x≠0),f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x};(x \neq 0), is continuous function at x=0x = 0, then f(0)f(0) equals

A

14\frac{1}{4}

B

−14- \frac{1}{4}

C

18\frac{1}{8}

D

−18- \frac{1}{8}

Answer

−18- \frac{1}{8}

Explanation

Solution

If f(x)f(x) is continuous at x=0,x = 0, then, f(0)=lim⁡x→0f(x)f(0) = \lim_{x \rightarrow 0}f(x)

=lim⁡x→02−x+4sin⁡2x= \lim_{x \rightarrow 0}\frac{2 - \sqrt{x + 4}}{\sin 2x} ,(00 form)\left( \frac{0}{0}\ \text{form} \right)

Using L–Hospital’s rule, f(0)=lim⁡x→0(−12x+4)2cos⁡2x=−18f(0) = \lim_{x \rightarrow 0}\frac{\left( - \frac{1}{2\sqrt{x + 4}} \right)}{2\cos 2x} = - \frac{1}{8}.