Question

If $\underset{x\to 0}{lim}{(\cos x+a\sin bx)}^{1/x}=e^2$ then the values of $a$ and $b$ are

• A. $a=1,b=2$
• B. $a=2,b=\frac{1}{2}$
• C. $a=2\sqrt{2},b=\frac{1}{\sqrt{2}}$
• D. $a=4,b=2$

Answer 1

Answer: (A), (C)

$a=1,b=2$
$a=2\sqrt{2},b=\frac{1}{\sqrt{2}}$

$\underset{x\to 0}{lim}{(\cos x+a\sin bx)}^{\frac{1}{x}}=e^2$ ( $1^{\infty }$ form)
$\Rightarrow exp\left(\underset{x\to 0}{lim}\frac{(\cos x+a\sin bx-1)}{x}\right)=e^2\Rightarrow exp\left(\underset{x\to 0}{lim}\frac{(-\sin x+a\cos bx)}{1}\right)=e^2\Rightarrow e^{ab}=e^2\Rightarrow ab=2$