Approximate the graph of $e^{x^2}$

Question

While solving the integrals related to $e^{x^2}$, we try to approximate it. My question is there some function of which the graph is approximately like that of $e^{x^2}$?

Answer 1

Using the power series representation we have

$$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}=1+x^2+\frac{x^4}{2!}+...$$

Taking more terms will give a better and better approximation.

Answer 2

The Taylor series for $e^{x^2}$ is $$1 + x^2 + \frac{x^4}{2} + O(x^6),$$ so that any function that you sum $1$ to an even power of $x$ will ressemble approximately the function you mention. Take a look at its behaviour in Wolfram Alpha.