Statement Calculate limx→0tanxx\displaystyle\lim_{x\to 0}\frac{\tan x}{x}x→0limxtanx. Solution It presents as a 0/00/00/0 form. Writing tanx=sinxcosx\tan x = \dfrac{\sin x}{\cos x}tanx=cosxsinx: tanxx=sinxx⋅1cosx.\frac{\tan x}{x} = \frac{\sin x}{x}\cdot\frac{1}{\cos x}.xtanx=xsinx⋅cosx1. As x→0x\to 0x→0: sinxx→1\dfrac{\sin x}{x}\to 1xsinx→1 (standard limit) and 1cosx→1\dfrac{1}{\cos x}\to 1cosx1→1. Hence limx→0tanxx=1⋅1=1.\lim_{x\to 0}\frac{\tan x}{x} = 1\cdot 1 = \boxed{1}.limx→0xtanx=1⋅1=1. Alternatively, using the equivalent functions tan(□)∼□\tan(\square)\sim\squaretan(□)∼□: tanxx∼xx=1\dfrac{\tan x}{x}\sim\dfrac{x}{x}=1xtanx∼xx=1. Links Topics: Limits Concepts: Equivalent functions · Standard limits Skills: Calculating limits Exercise type: Limit calculation