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