if `f(x)=(e^(x^(2))-cosx)/(x^(2))`, for `xne0` is continuous at `x=0`, then value of f(0) is

Explanation

`underset(x to 0) limf(x)=underset(x to0)lim(e^(x^(3))-cosx)/(x^(2))`
`=underset(x to 0)lim((e^(x^(2))-1)+(1-cosx))/(x^(2))`
`=(underset(x to 0)lim(e^(x^(2))-1)/(x^(2)))+(underset(x to 0)lim(1-cos x)/(x^(2)))=L_(1)+L_(2)" "...`(Say)
In `L_(1) Put x^(2)=y. ` Then as `x to 0, y to 0.`
`thereforeL_(1)=underset(x to 0)lim(e ^(x^(2)-1))/(x^(2))=underset(y to 0) lim(e^(y)-1)/(y)=1`
`L_(2)=underset(x to 0)lim(1-cos x)/(x^(2))=underset(x to 0)lim (1-cosx)/(x^(2))*(1+cosx)/(1+cosx)`
`=underset( x to 0)lim (1-cos ^(2)x)/(x^(2)(1+cos x))=underset(x to 0) lim(sin^(2)x)/(x^(2))*(1)/(1+cso x)`
`=(underset( x to 0)lim(sin x)/(x))^(2)*(underset(x to 0)lim (1)/(1+cos x))=1^(2)xx(1)/(1+1)=1/2`
`thereforeunderset(x to 0)lim f(x)=1+1/2=3/2`
Now, f is continous at `x=0`
`therefore f(0) =underset(x to 0)lim f(x) therefore f(0)=3/2*`