# Given the rational function `f(x)=(x^2 - 5) /( x +3)` find a. the horizontal asymptotes, verticle and obliquemust show work

lemjay | Certified Educator

`f(x)= (x^2-5)/(x+3)`

> To determine if the rational function has a vertical asymptote, let's consider the expression in the denominator. Note that a zero denominator is not allowed. If there are values of x that makes the denominator zero, then the rational function has vertical asymptotes.

To solve for the vertical asymptote of the given function, set the denominator equal to zero.

`x+3=0`

`x=-3`

Hence, the function has a vertical asymptote at x=-3.

> To determine if the rational function has a horizontal or oblique asymptote, compare the degree of the numerator with the denominator.

Since the degree of the numerator of the given function is higher than the denominator, then f(x) has an oblique asymptote.

To solve for the oblique asymptote, divide the numerator by the denominator using long division.

`x` `-` `3`

`x+3` `|bar(x^2+0x-5)`

`(-)` `x^2+3x`

`--------`

`-3x-5`

`(-)`  `-3x-9`

`--------`

`4`

Thus, the slant asymptote of f(x) is the line `y=x-3` .

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Answer: Vertical Asymptote of f(x): `x=-3`

Oblique Asymptote of f(x):  `y=x-3`