Question

Not using the L'Hopital's rule, or graph/table........ I need to find the limit of tan4x/tan3x as x approaches 0.

Answer 1

$\lim\limits_{x\to0}\dfrac{\tan4x}{\tan3x}\stackrel{[H]}=\lim\limits_{x\to0}\left[\left(\dfrac{1}{\cos^24x}\cdot4\right):\left(\dfrac{1}{\cos^23x}\cdot3\right)\right]\\\\=\lim\limits_{x\to0}\left(\dfrac{4}{\cos^24x}\cdot\dfrac{\cos^23x}{3}\right)=\dfrac{4}{1}\cdot\dfrac{1}{3}=\dfrac{4}{3}\\------------------\\(\tan x)'=\dfrac{1}{\cos^2x}\\\\\lim\limits_{x\to0}\cos x=\cos0=1$