Question

How do you determine the limit of `(x² - 3x - 2)/(x² - 5)` as x approaches infinity?

Answer 1

`lim_(xrarrinfty)(x^2-3x-2)/(x^2-5)=1`

Explanation:

Given,

`lim_(xrarrinfty)(x^2-3x-2)/(x^2-5)`

Divide every term by the term with the highest degree in the denominator.

`=lim_(xrarrinfty)(x^2/color(blue)(x^2)-(3x)/color(blue)(x^2)-2/color(blue)(x^2))/(color(blue)(x^2)/color(blue)(x^2)-5/color(blue)(x^2))`

`=lim_(xrarrinfty)(1-3/x-2/x^2)/(1-5/x^2)`

Substitute `x=infty`.

`=lim_(xrarrinfty)(1-3/infty-2/infty^2)/(1-5/infty^2)`

Any constant divided by `infty` has a limit of `0` since the quotient would approach `0`.

`=lim_(xrarrinfty)(1-0-0)/(1-0)`

`=lim_(xrarrinfty)1/1`

`=1`