Answer to Question #299118 in Real Analysis for Dhruv bartwal

Question

Find

lim tan x sec^2 x- x/x^3

x→0

Answer 1

$$\lim\limits_{x\to0}\dfrac{\tan x\sec^2 x-x}{x^3}=\lim\limits_{x\to0}\dfrac{(\tan x\sec^2 x-x)'}{(x^3)'}$$

$$=\lim\limits_{x\to0}\dfrac{\sec^4 x+2\tan^2 x\sec^2 x-1}{3x^2}$$

$$=\lim\limits_{x\to0}\dfrac{\sec^2 x(1+\tan ^2 x)+2\tan^2 x\sec^2 x-1}{3x^2}$$

$$=\lim\limits_{x\to0}\dfrac{\tan ^2 x+3\tan^2 x\sec^2 x}{3x^2}$$

$$=\dfrac{1}{3}\lim\limits_{x\to0}\dfrac{\tan ^2 x}{x^2}+\dfrac{3}{3}\lim\limits_{x\to0}\dfrac{\tan ^2 x}{x^2}(\lim\limits_{x\to0}\sec^2 x)$$

$$=\dfrac{1}{3}(1)+1(1)=\dfrac{4}{3}$$