lim (x -> 0) sin(x)/(x+tan(x))
Can someone show me step by step method of solving this. I tried it, but my answer doesn't match the one in the book.
It is commonly known that $\displaystyle \tan(x)\sim{x}$ and $\displaystyle \sin(x)\sim{x}$
$\displaystyle \therefore\lim_{x\to{0}}\frac{\sin(x)}{x+\tan(x)}= \lim_{x\to{0}}\frac{x}{2x}=\frac{1}{2}$
Alternatively
you can divide by x to get
$\displaystyle \lim_{x\to{0}}\frac{\frac{\sin(x)}{x}}{1+\frac{\ta n(x)}{x}}$
Evaluating each separately we get
$\displaystyle \lim_{x\to{0}}\frac{\sin(x)}{x}=\lim_{x\to{0}}\fra c{\sin(x)-\sin(0)}{x-0}=\bigg(\sin(x)\bigg)'\bigg|_{x=0}=1$
and $\displaystyle \lim_{x\to{0}}\frac{\tan(x)}{x}=\lim_{x\to{0}}\fra c{\tan(x)-\tan(0)}{x-0}=\bigg(\tan(x)\bigg)'\bigg|_{x=0}=1$
Thus we have
$\displaystyle \frac{1}{1+1}=\frac{1}{2}$
also you could utilize power series but that might be too advanced