Question

lim x rightarrow 0 1 - cos ( x2 ) / 1 - cosx The limit has to be evaluated without using l'Hospital'sRule.

Answer 1

Answer with Step-by-step explanation:

Given

`f(x)=(1-cos(2x))/(1-cos(x))\\\\\lim_(x \rightarrow 0)f(x)=\lim_(x\rightarrow 0)(\frac{1-(cos^2{x}-sin^2{x})}{1-cos(x)})\\\\(\because cos(2x)=cos^2x-sin^2x)\\\\\lim_(x \rightarrow 0)f(x)=\lim_(x\rightarrow 0)((1-cos^2x)/(1-cos(x))+(sin^2x)/(1-cosx))\\\\=\lim_(x\rightarrow 0)(((1-cosx)(1+cosx))/(1-cosx)+(sin^2x)/(1-cosx))\\\\=\lim_(x\rightarrow 0)((1+cosx)+(sin^2x)/(1-cosx))\\\\\therefore \lim_(x \rightarrow 0)f(x)=1`