Question

Evaluate the Given limit: $\lim_{x\rightarrow 0}$ $\frac{ax+xcos\,x}{b\,sin\,x}$

Answer 1

$\lim_{x\rightarrow 0} \frac{ax+xcos\,x}{b\,sin\,x}$
At x = 0, the value of the given function takes the form 0/0.
$\lim_{x\rightarrow 0}$ $\frac{cos^2x-1}{cos\,x-1}$ =\frac{1}{b}$$\lim_{x\rightarrow 0} $\frac{x(a+cos\,x)}{sin\,x}$
=\frac{1}{b}$$\times$$\lim_{x\rightarrow 0}($\frac{x}{sin\,x}$) \times$$\lim_{x\rightarrow 0} (a + cosx)
=$\frac{1}{b}$ $\times$ (\lim_{x\rightarrow 0}$$\frac{1}{\frac{sin\,x}{x}}) $\times$ $\lim_{x\rightarrow 0}$ (a + cosx)
= $\frac{1}{b}$ $\times$ (a + cos0) [$\lim_{x\rightarrow 0}$ $\frac{sin\,x}{x}$ = 1]
= $a+\frac{1}{b}$