Evaluate: `("lim")_(xvecoo)((int0xe^x^2dx)^2)/(int0x e^(2x)^2dx)`

Explanation

Since `e^(x^(2))gt0,e^(2x^(2))gt0` in `[0,x]` where `xgt0`,
`int_(0)^(x)e^(x^(2))dx` and `int_(0)^(2x^(2))dxto oo` as `x to oo`
`L=lim_(xto oo)((int_(0)^(x)e^(x^(2)))^(2))/(int_(0)^(x)e^(2x^(2)))` is of the form `(oo)/(oo)`
Therefore using L'Hopital's rule
`L=lim_(xto oo) (2e^(x^(2))int_(0)^(x)e^(x^(2))dx)/(e^(2x^(2)))`
`=2lim_(xto oo) (int_(0)^(x)e^(x^(2))dx)/(e^(x^(2)))`
`2lim_(xto oo) (e^(x^(2)))/(2x^(x^(2)))`
`=lim_(xto oo)1/x=0`