How do you find the asymptotes for #f(x) = (x+3)/(x^2 + 8x + 15)#?

Question

Serenity Jones · Accepted Answer

Only one vertical asymptote is #x+5=0#. There is no horizontal or slanting asymptote.

To find all the asymptotes for function #y=(x+3)/(x^2+8x+15)#, let us first start with vertical asymptotes, which are given by putting denominator equal to zero or #x^2+8x+15=0#, which when factorized becomes #(x+3)(x+5)=0# i.e. #x+3=0# and #x+5=0#. But we have #(x+3)# in numerator too. Hence, function becomes #y=1/((x+5)# and hence only one vertical asymptote #y=x+5#. There is no horizontal (which requires degree of numerator and denominator to be equal) and slanting asymptote (which requires degree of numerator to be just greater than that of denominator by #1#). graph{(x+3)/(x^2+8x+15) [-10, 10, -5, 5]}

Madelyn Anderson · Answer

undefined To find the asymptotes for the function $f(x) = \frac{x+3}{x^2 + 8x + 15}$, follow these steps:

1. Determine if there are any vertical asymptotes by finding the values of $x$ for which the denominator is zero. In this case, solve $x^2 + 8x + 15 = 0$ to find the roots.
2. Factor the denominator, if possible, and then find the roots.
3. If there are any horizontal asymptotes, determine them by examining the behavior of the function as $x$ approaches positive or negative infinity.
4. If there are any slant (oblique) asymptotes, use polynomial division or long division to divide the numerator by the denominator. The quotient obtained will represent the slant asymptote.

After following these steps, you'll have the vertical, horizontal, and slant asymptotes, if they exist, for the function $f(x)$.