`lim_(x to oo) ((x^(2)+ 5x+3)/(x^(2)+x+3))^(x)` is equal to

Options

A. `e^(4)`
B. `e^(2)`
C. `e^(3)`
D. e

Answer

A

Explanation

We have ` lim_( x to oo) ((x^(2)+5x +3)/(x^(2)+x+3))^(x)`
`Rightarrow e^(lim_(x to oo)) ((x^(2) +5x+3)/(x^(2) +x+3)-1)x`
` Rightarrow e^(lim_(x to oo) ((x^(2) +5x +3 -x^(2)-x-3)/(x^(2) +x+3))x`
` Rightarrow e^(lim_(x to oo)) ((4x^(2))/(x^(2) +x+3)) =e^(4)`