Calculus 1A — Limits & Continuity

Limits at infinity & asymptotes

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: sometimes we ask where a function heads not toward a point but toward the far horizon — as x runs off to infinity. The answer is a flat line the curve snuggles up to forever: a horizontal asymptote.

For a ratio of polynomials, only the highest powers matter way out there — everything else is a rounding error. So (3x² + 2) / (x² − 5) heads to 3/1 = 3: the x² terms dominate and their coefficients decide the destination.

rat\left(x\right) = \frac{3 \cdot x^{2} + 2}{x^{2} - 5}

a ratio of quadratics

far_{out} = \mathrm{rat}\left(1000000\right)

3.0000

a million out → essentially 3

✓ pass

\mathrm{abs}\left(far_{out} - 3\right) < 0.001

leading coefficients 3 and 1 win the tug-of-war

rat(x)

the curve flattens onto the horizontal asymptote y = 3

Real-world hook: horizontal asymptotes are limits in disguise everywhere — a falling object's terminal velocity, a medication's steady blood concentration, a website's response time as traffic approaches the server's capacity. Each levels off at a value it approaches but never quite reaches.

Try it yourself: what does (5x² − 2x) / (2x² + 7) approach as x → ∞?

Show the answer

Leading coefficients 5 and 2 ⇒ the limit is 5/2 = 2.5.