proof of lim tanx/x x-0

Feb 2010
460
37
tan(x) / x as x approaches 0 proof.i tried the expansion and lhospital rule and got it.is there a third method to do it maybe by graph
 
Mar 2010
715
381
\(\displaystyle \lim_{x\to 0}\bigg\{\dfrac{\tan{x}}{{x}}\bigg\} = \lim_{x\to 0}\bigg\{\frac{\left(\frac{\sin{x}}{\cos{x}}\right)}{x}\bigg\} = \lim_{x\to 0}\bigg\{\left(\dfrac{\sin{x}}{{x}}\right)\left(\dfrac{1}{\cos{x}}\right)\bigg\} = \) \(\displaystyle \lim_{x\to 0}\bigg(\dfrac{\sin{x}}{{x}}\bigg)\lim_{x\to 0}\bigg(\dfrac{1}{{\cos{x}}}\bigg) = 1.\)