The value of `lim_(xrarroo) (e^(sqrt(x^(4)+))-e^((x^(2)+1)))` is

Options

A. 0
B. e
C. 1/e
D. `-oo`

Answer

D

Explanation

`L=underset(xrarroo)(lim)(e^((sqrt(x^(4)+1)))-e^((x^(2)+1)))`
`=underset(xrarroo)(lim)e^(x^(2)+1)[e^(sqrt(x^(4)+1)-(x^(2)+1))-1]`
Now, `underset(xrarroo)(lim)[sqrt(x^(4)+1)-(x^(2)+1)]`
`=underset(xrarroo)(lim)(x^(4)+1-(x^(4)+1+2x^(2)))/(sqrt(x^(4)+1)+(x^(2)+1))`
`=underset(xrarroo)(lim)(-2x^(2))/(x^(2)(sqrt(1+(1//x^(4)))+1+(1//x^(2))))`
` =-1`
`therefore" "L=ooxx((1)/(e)-1)`
`=-oo`