What does a function do as x grows without bound? Its end behaviour.
Try this
1
(3x² + 1) / (x² + 5) = 0.67
How to find limits at infinity
For a ratio of polynomials, look at the highest power on top and bottom. Same degree ⇒ limit is the ratio of leading coefficients. Bigger degree on top ⇒ limit is ±∞. Bigger degree on bottom ⇒ limit is 0.
Your turn
Find lim (x→∞) (2x³ − x) / (5x³ + 7).
Watch out
Don't just plug in ∞ and write ∞/∞ — that's undefined. Compare the growth rates (highest powers) instead, or divide through by the dominant term.
Try it
lim (x→∞) (3x² + 1)/(x² + 5)
Both top and bottom dominated by x². Ratio → 3/1 = 3.