How do you find the horizontal asymptote for #f(x) = (x+3)/(x-2)^3 #?

Question

James Allen · Accepted Answer

Look at the coefficients on the variables with the greatest exponents for both the numerator and denominator ...

Numerator : highest exponent is 1 #(1x^1)#and the coefficient is also 1. Denominator : highest exponent is 3 #(1x^3)# and the coefficient is 1 So we have #(1x)/(1x^3)#. As #xrarroo#, this rational function goes to zero . Therefore, horizontal asymptote is #y=0# hope that helped

Sophie Adams · Answer

undefined To find the horizontal asymptote for $ f(x) = \frac{x+3}{(x-2)^3} $, we need to examine the behavior of the function as $ x $ approaches positive or negative infinity.

When $ x $ approaches positive or negative infinity, the highest power terms in the numerator and denominator dominate the function's behavior. In this case, the highest power terms in the denominator are $ (x-2)^3 $.

Since the denominator is raised to the power of 3, the degree of the denominator is greater than the degree of the numerator (which is 1). When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at $ y = 0 $.

Therefore, the horizontal asymptote for the function $ f(x) = \frac{x+3}{(x-2)^3} $ is $ y = 0 $.