How do you find the vertical, horizontal or slant asymptotes for # f(x)=(1-5x)/( 1+2x#?

Question

Benjamin Achtenberg · Accepted Answer

#y = -(5)/(2)#

#x = -1/2# Firstly, find what #x# cannot be for the vertical asymptotes So the denominator of the fraction cannot be equal to #0#. #1+2x != 0# #x!= -0.5# Meaning that #x = -0.5# is the vertical asymptote. Then, find the horizontal/slant asymptotes by only looking at the highest degree of #x#. #f(x) = (1-5x)/(1+2x)# As #x rarr oo#, we only care about the values that will change the function most, so we can just look at the highest degree of #x#. Thus, he asymptote will be: #y = (-5x)/(2x)# #y = -(5)/(2)#

Caleb Alexander · Answer

undefined To find the vertical, horizontal, or slant asymptotes of the function $ f(x) = \frac{1 - 5x}{1 + 2x} $, follow these steps:

1. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator doesn't. Set the denominator $ 1 + 2x $ equal to zero and solve for $ x $. The values of $ x $ obtained are the vertical asymptotes, if any.

2. **Horizontal Asymptotes**: Horizontal asymptotes occur when $ x $ approaches positive or negative infinity. To find horizontal asymptotes:
   - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
   - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $.
   - If the degrees of both the numerator and denominator are equal, divide the leading coefficient of the numerator by the leading coefficient of the denominator to find the horizontal asymptote.

3. **Slant Asymptotes**: Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. To find a slant asymptote, perform polynomial long division to divide the numerator by the denominator. The quotient obtained represents the equation of the slant asymptote.

By following these steps, you can determine the vertical, horizontal, or slant asymptotes of the function $ f(x) = \frac{1 - 5x}{1 + 2x} $.