proof of lim tanx/x x-0

**prasum** · May 25th 2010, 11:44 PM

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

**TheCoffeeMachine** · May 26th 2010, 12:24 AM

$\lim_{x\to 0}\bigg\{\dfrac{\tan{x}}{{x}}\bigg\} = \lim_{x\to 0}\bigg\{\frac{\left(\frac{\sin{x}}{\cos{x}}\right )}{x}\bigg\} = \lim_{x\to 0}\bigg\{\left(\dfrac{\sin{x}}{{x}}\right)\left(\d frac{1}{\cos{x}}\right)\bigg\} =$ $\lim_{x\to 0}\bigg(\dfrac{\sin{x}}{{x}}\bigg)\lim_{x\to 0}\bigg(\dfrac{1}{{\cos{x}}}\bigg) = 1.$

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