If $f(x)=\frac{e^{x^2}-\cos x}{x^2}$ if $x
eq 0$ is continuous at $x=0$, then $f(0)=$.

Question

If $f(x)=\frac{e^{x^2}-\cos x}{x^2}$ if $x 
eq 0$ is continuous at $x=0$, then $f(0)=$.

Aicharya · Accepted Answer

For continuity at $x=0, \lim _{x ightarrow 0} f(x)=f(0)$ $\begin{aligned} & \Rightarrow f(0)=\lim _{x ightarrow 0} \frac{e^{x^2}-\cos x}{x^2}=\lim _{x ightarrow 0} \ & \frac{\left(1+x^2+\frac{x^4}{2 !}+\ldots \ldotsight)-\left(1-\frac{x^2}{2 !}+\ldots . . .ight)}{x^2} \ & =\lim _{x ightarrow 0} \frac{\frac{3}{2} x^2+\ldots . .}{x^2}=\frac{3}{2} \end{aligned}$

If $f(x)=\frac{e^{x^2}-\cos x}{x^2}$ if $x \neq 0$ is continuous at $x=0$, then $f(0)=$.

Solution