Find the Asymptotes y=(x^3)/(x^2+4)

Problem

y=(x3)/(x2+4)

Solution

1. Identify vertical asymptotes by finding values of x that make the denominator zero. Since x2+4=0 has no real solutions (x2=−4, there are no vertical asymptotes.

2. Compare the degrees of the numerator and denominator to find horizontal or slant asymptotes. The degree of the numerator is 3 and the degree of the denominator is 2 Since the numerator's degree is exactly one higher than the denominator's, there is a slant (oblique) asymptote.

3. Perform polynomial long division to find the equation of the slant asymptote. Divide x3 by x2+4

x3=x*(x2+4)−4*x

4. Rewrite the function using the result of the division to separate the linear part from the remainder.

y=x−(4*x)/(x2+4)

5. Determine the limit as x approaches infinity. As x→∞ the term (4*x)/(x2+4) approaches 0 The slant asymptote is the linear part of the quotient.

Final Answer

Slant Asymptote: *y=x

Want more problems? Check here!