Question

1. Find the limit as x approaches 0 , of the function (1- cos x)/x^2. (5 mks)

2.find the limit as x approaches y, of the function (((sin^2 x) -(sin^2 y))/(x^2 -y^2)). 6mks

Answer 1

`\displaystyle\lim_(x\to0)(1-\cos x)/(x^2)=\lim_(x\to0)(1-\cos^2x)/(x^2(1+\cos x))=\lim_(x\to0)(\sin^2x)/(x^2(1+\cos x))`

`=\displaystyle\lim_(x\to0)(\sin^2x)/(x^2)\cdot\lim_(x\to0)\frac1{1+\cos x}=\left(\lim_(x\to0)\frac{\sin x}x\right)^2\lim_(x\to0)\frac1{1+\cos x}=1^2\cdot\frac1{1+1}=\frac12`