What is the value of `lim_(xtooo) ((x-2)/(x+2))^(x+2)` = ?

Options

A. 0
B. `e^(4)`
C. `e^(-2)`
D. `e^(-4)`

Answer

D

Explanation

`underset(xto0)lim((x-2)/(x+2))^(x+2)`
can be written as
`underset(xto0)lim((x+2-4)/(x+2))^(x+2)`
or, `underset(xtooo)lim{1-(4)/((x+2))}^(x+2)`
Putting x+2=t
when `xtooo,t""tooo` '
So, `underset(t""tooo)lim{1-(4)/(t)}^(t)`
or, `underset(t""tooo)lim{1-(4)/(t)}^((t)/(4)xx4)`
or, `underset(t""tooo)lim{(1-(4)/(t))^((t)/(4))}^(4)=(e^(-1))^(4)=e^(-4)`