Question

If `f(x) =(e^(x^2)-cosx)/x^2`, for x= 0, is continuous at x = 0, find f(0).

Answer 1

f(x) is continuous at x = 0

`lim_(x->0)f(x)=f(0)`

`f(0)=lim_(x->0)f(x)=lim_(x->0)(e^(x^2)-cosx)/x^2=lim_(x->0)((e^(x^2)-1)+(1-cosx))/x^2`

`=lim_(x->0)((e^(x^2)-1)/x^2+(1-cosx)/x^2)`

`=lim_(x->0)((e^(x^2)-1)/x^2+(2sin^2(x/2))/x^2)`

`=lim_(x->0)((e^(x^2)-1)/x^2+2(sin(x/2)/x)^2)`

`=lim_(x->0)((e^(x^2)-1)/x^2+2(sin(x/2)/(x/2)xx1/2)^2)`

`=lim_(x->0)(e^(x^2)-1)/x^2+1/2(lim_(x->0)sin(x/2)/x)^2`

`=1+1/2(1)^2`

`=3/2`

Thus,f(0)=3/2