How do you find the asymptotes for #(x^4 - 2x + 3) / (6 - 5x^3)#?

Question

Sophie Adams · Accepted Answer

There is an asymptote where# x =root3 (6/5)# How many 0.1 go into 1. Answer 10 How many 0.01 go into 1 answer 100 As the divisor approaches zero the answer approaches infinity.- an asymptote When #6-5x^3=0# there is an asymptote hence the answer

Lily Anderson · Answer

undefined To find the asymptotes for $ \frac{{x^4 - 2x + 3}}{{6 - 5x^3}} $, we examine both horizontal and vertical asymptotes.

1. Vertical Asymptotes:
   Vertical asymptotes occur where the denominator of the rational function equals zero. Set the denominator $ 6 - 5x^3 $ equal to zero and solve for $ x $:
   $$ 6 - 5x^3 = 0 $$
   $$ 5x^3 = 6 $$
   $$ x^3 = \frac{6}{5} $$
   $$ x = \sqrt[3]{\frac{6}{5}} $$

Therefore, there is a vertical asymptote at $ x = \sqrt[3]{\frac{6}{5}} $.

2. Horizontal Asymptotes:
   To find horizontal asymptotes, we compare the degrees of the numerator and the denominator. Since the degree of the numerator (4) is greater than the degree of the denominator (3), there is no horizontal asymptote.

Thus, the asymptotes for the given rational function are:

- Vertical asymptote: $ x = \sqrt[3]{\frac{6}{5}} $
- There is no horizontal asymptote.