Find the limit of the rational function ​

Question

Find the limit of the rational function ​(a) as x→∞ and​ (b) as x→−∞.

lim (5x⁸+2x³-1 / 2x⁶-9x+2)

Answer 1

Since the degree of the numerator (8) is greater then the degree of the denominator (6) there is no horizontal asymptote,. Since the degree of the numerator and denominator are both even, and the leading coefficients are both positive, the end-behavior is "up-up". That means limit_x->-∞ = +∞.

To visualize suppose x is something like - 1000.

f(-1000) = 5(-1000)⁸/2(-1000)⁶. These numbers just keep getting bigger and bigger.

By the way if the degree of the numerator was 6 (instead of 8) the limit would be 5/2.

See this graph to get a better idea:

desmos.com/calculator/9w7u5xaz1h