Question

Determine the end behavior of each rational function below.
f(x) = 7x3 + 2 / 4x5 − 3x + 1

Answer 1

When x ⇒ +∞ and x ⇒ -∞, f(x) ⇒ 0 from the positive region.

Explanation:

Hi there!

Let´s write the function:

f(x) = (7x³ + 2) / (4x⁵ - 3x + 1)

When x ⇒ +∞ we can neglect the constant values (2 and 1). So:

f(x) ⇒ 7x³ /(4x⁵ - 3x)

f(x) ⇒ 7x³ / x(4x⁴ - 3)

f(x) ⇒ 7x² / 4x⁴

f(x) ⇒ 7/4x² ⇒ 0 from the positive region

when x ⇒ -∞

f(x) ⇒ 7/ 4x² ⇒ 0 from the positive region because x² is always positive.

Then when x is a very big number (positive or negative), f(x) will be approximately 0 and positive.