Identify the vertical asymptotes of f (x) = quantity x minus 4 over quantity x squared plus 13 x plus 36

Expression: f (x) = [x - 4] / [x^2 + 13x + 36].

The vertical asympotes is f (a) when the denominator of f (x) is zero and at least one side limit when you approach to a is infinite or negative infinite.

The we have to factor the polynomial in the denominator to identify the roots and the limit of the function when x approachs to the roots.

x^2 + 13x + 36 = (x + 9) (x + 4) = > roots are x = - 9 and x = - 4

Now you can write the expresion as: f (x) = [x - 4] / [ (x + 4) (x+9) ]

Find the limits when x approachs to each root.

Limit of f (x) when x approachs to - 4 by the right is negative infinite and limit when x approach - 4 by the left is infinite, then x = - 4 is a vertical asymptote.

Limit of f (x) when x approachs to - 9 by the left is negative infinite and limit when x approach - 9 by the right is infinite, then x = - 9 is a vertical asymptote.

Answer: x = - 9 and x = - 4 are the two asymptotes.