How do you find the asymptotes for #[(3x^2) + 14x + 4] / [x+2]#?

Question

Elias Ackerman · Accepted Answer

One vertical asymptote #x+2=0# and one slanting asymptote given by #y=3x#.

To find the asymptotes of #(3x^2+14x+4)/(x+2)# we first observe the denominator #(x+2)#, which shows that the function has one vertical asymptote #x+2=0#. Further, highest degree of numerator is two and that of denominator is one, and their ratio is #3x^2/x=3x#. Hence, we have a slanting asymptote given by #y=3x# graph{(3x^2+14x+4)/(x+2) [-10, 10, -5, 5]}

Elijah Allen · Answer

undefined To find the asymptotes of the rational function $ \frac{3x^2 + 14x + 4}{x + 2} $, you need to check for vertical and horizontal asymptotes.

Vertical asymptotes occur where the denominator is equal to zero, but the numerator is not. So, set $ x + 2 = 0 $ and solve for $ x $ to find any vertical asymptotes.

Horizontal asymptotes occur when the degree of the numerator and the degree of the denominator are the same. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

If the degree of the denominator is greater than the degree of the numerator by one, the horizontal asymptote is at $ y = 0 $.

If the degree of the denominator is greater than the degree of the numerator by more than one, there is no horizontal asymptote.

Evaluate the limits as $ x $ approaches positive or negative infinity to verify the horizontal asymptotes, if any.