How to find the asymptotes of #f(x) = (x+3)(x-2)^3 # ?

Question

Madelyn Anderson · Accepted Answer

There are no asymptotes for this one.

Vertical asymptotes occur mainly where a variable is undefined at a particular value. Since the domain is #(-oo,oo)#, that won't be the case. Horizontal asymptotes occur when the value of the function approaches a fixed number as x approaches #oo# and #-oo#. At huge positive values of x, f(x) will approach #oo#. The same at large negative values of x. Since the conditions for neither vertical nor horizontal asymptotes are met, there are no asymptotes here.

Serenity Jones · Answer

undefined To find the asymptotes of the function $ f(x) = (x + 3)(x - 2)^3 $, we need to consider vertical asymptotes and horizontal asymptotes separately.

Vertical asymptotes occur where the function approaches infinity or negative infinity as $ x $ approaches a certain value. They typically arise from the denominator of a rational function being equal to zero.

Horizontal asymptotes occur when $ x $ approaches positive or negative infinity, and the function approaches a constant value.

1. Vertical Asymptotes:
   Set the factors of the function that can make the denominator zero equal to zero and solve for $ x $.
   In this case, we only have one factor that can make the function undefined, which is $ x - 2 = 0 $. Solving for $ x $, we get $ x = 2 $.
   Therefore, there is a vertical asymptote at $ x = 2 $.

2. Horizontal Asymptotes:
   To find horizontal asymptotes, analyze the behavior of the function as $ x $ approaches positive or negative infinity.
   Since the highest power of $ x $ in the function is $ x^3 $, and there are no terms in the function that can offset its effect, the function will behave like $ x^3 $ for large values of $ x $.
   Therefore, there is no horizontal asymptote because as $ x $ approaches infinity, $ f(x) $ will also approach infinity.

In summary, the asymptotes of $ f(x) = (x + 3)(x - 2)^3 $ are:
- Vertical asymptote at $ x = 2 $
- No horizontal asymptote.