How do you find the asymptotes for #y=(x^3 +3x^2-4x-10)/(x^2-4)#?

1 Answer
Jun 1, 2018

Answer:

Consider behaviour in the infinite domain limits and when the denominator goes to zero. The function has three asymptotic lines, #y=x#, #x=-2#, and #x=+2#.

Explanation:

In the limit as #xrarroo#, #yrarr x^3/x^2=x#.
In the limit as #xrarr-oo#, also #yrarr x^3/x^2=x#. So in both infinite domain limits the function tends to the diagonal straight line #y=x#, an asymptote in both cases.

The function has two poles on the real line, at #x=+-2#, where the denominator is zero and the numerator is non-zero. So the two lines #x=+-2# are vertical asymptotes of the functions.