How do you find all the asymptotes for # (4x)/(x-3) #?

Question

How do you find all the asymptotes for # (4x)/(x-3)   #?

Christian Antunez · Accepted Answer

Vertical asymptote: #x=3# & Horizontal asymptote: #y=4#

Given function:

#f(x)={4x}/{x-3}#

Setting #x-3=0\implies x=3#

The given function is undefined at #x=3# or #x=3# is a point of discontinuity.

Hence, the given curve has a vertical asymptote: #x=3#

Now, the horizontal asymptote:

#y=\lim_{x\to \pm\infty}f(x)#

#y=\lim_{x\to \pm\infty}(\frac{4x}{x-3})#

#y=4#

Serenity Johnson · Answer

undefined To find the asymptotes of the function (4x)/(x - 3), you need to consider two types of asymptotes: vertical and horizontal.

1. Vertical Asymptote: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. In this case, the vertical asymptote occurs at x = 3 because if you plug in x = 3 into the denominator, you get 0.

2. Horizontal Asymptote: To find horizontal asymptotes, examine the behavior of the function as x approaches positive or negative infinity. You can use the concept of limits. When x approaches infinity, the terms with the highest power dominate the function. So, for the given function, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and denominator. In this case, the horizontal asymptote is y = 4/1 = 4.

Therefore, the asymptotes for the function (4x)/(x - 3) are:
- Vertical asymptote: x = 3
- Horizontal asymptote: y = 4