If the function `f(x)=(e^(x^2)-cos x)/(x^2) ` for `x!=0` is continuous at `x=0` then `f(0)`

Explanation

Since f(x) is continousat `x=0`, we get
`f(0)lim_(x to 0)f(x)`
`:. f(0)lim_(x to 0) (e^(x^(2))-cos x)/(x^(2))`
=lim_(x to 0)((e^(x^(2))-1)+(1-cosx))/(x^(2))`
`lim_(x to 0)((e^(x^(2))-1)+2-"sin"^(2)(x)/(2))/(x^(2))`
`lim _(x to 0) (e^(x^(2))-1)/(x^(2))+lim_(x to 0)(2"sin"(x)/(2))/(x^(2))`
`=1+2* lim_(x to 0)("sin"^(2)(x)/(2))/(((x)/(2))^(2)xx(1)/(4))`
`=1+2*lim_(x to 0) (("sin"(x)/(2))/((1)/(2)))^(2)xx(1)/(4)`
`=1+2xx1xx(1)/(4)=1+(1)/(2)=(3)//(2)`
`:. f(0)=(3)/(2)`