How do you find the vertical, horizontal and slant asymptotes of: # f(x)=(1/(x-10))+(1/(x-20))#?

Question

Serenity Johnson · Accepted Answer

Vertical asymptotes: #x=20#.

Horizontal asymptotes: #y=0#

Slant asymptotes: none.

Vertical asymptotes: you have to look for the points in which the function is not defined. The way the function is written, it is really easy to see that the first denominator vanished for #x=10#, and the second for #x=20#. Horizontal asymptotes: you have to consider the limit as #x# approaches #\pm\infty#. Since in both cases you have a behaviour like #1/(\pm\infty)#, the horizontal aymptote is the #x# axis, i.e. #y=0# Slant asymptotes: since you have horizontal asymptotes, you can't have slant asymptotes.

Lily Anderson · Answer

undefined To find the vertical asymptotes, identify the values of $x$ that make the denominator of any fraction in the function equal to zero. In this case, the vertical asymptotes occur at $x = 10$ and $x = 20$.

Horizontal asymptotes are determined by the behavior of the function as $x$ approaches positive or negative infinity. If the degrees of the numerator and denominator of the function are the same, the horizontal asymptote is the ratio of the leading coefficients. In this case, both terms in the function have a degree of 1, so the horizontal asymptote can be found by comparing the leading coefficients, which are both 1. Therefore, there is a horizontal asymptote at $y = 0$.

Slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the degrees are the same, so there are no slant asymptotes.