Calculus I › Limits › full formula sheet
Limits at infinity
Where the graph settles as x runs away — and how to read it off in seconds.
On this page x → ∞ (and x → −∞) replaces x → a. The limit laws still apply — but “infinity” is a direction, never a number you can do arithmetic with.
Notation in this lesson
- limx→∞
- x grows without bound
- y = L
- horizontal asymptote — the line the graph settles toward
- 1/xn → 0
- the terms that die as x → ∞
Where it comes from
The problem: end behavior — where does the graph go as x gets huge? For rational functions this decides horizontal asymptotes, and the naive move is catastrophic:
Three “infinity over infinity” limits, three different answers. ∞/∞ is indeterminate — the ratio of growth rates decides, never a fixed rule. The technique that resolves all three at once: divide by the highest power, turning every term into something that → 0 except the dominant balance.
Derivation
The definition swaps δ for N (“eventually”), then we prove lim(x→∞) 1/x = 0 — the engine inside every divide-by-highest-power computation.
Key steps shown; the argument above is complete. For x → −infinity the definition reads x < N (N very negative) ⇒ |f−L| < ε. And 1/xn → 0 for every n > 0 by the same argument — which is why dividing by the highest power kills every non-dominant term.
How to use it
The procedure for rational functions, every time:
- Divide every term by the highest power of x in the denominator.
- Let x → ∞: every 1/xn term → 0. Only the dominant balance survives.
- Read the degree shortcut: deg(top) < deg(bottom) → 0 · deg(top) = deg(bottom) → ratio of leading coefficients · deg(top) > deg(bottom) → ±infinity (no horizontal asymptote).
- Check x → −infinity separately when degrees differ — the sign can flip.
Horizontal asymptotes
lim(x→∞) f(x) = L means the line y = L is a horizontal asymptote — the graph settles toward it. The two ends can have different asymptotes (arctan x → π/2 at +infinity, −π/2 at −infinity), and a graph may cross its horizontal asymptote — even infinitely often, like (sin x)/x. The asymptote describes the tail, not a barrier.
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — equal degrees: lim(x→∞) (3x²+1)/(x²−5)
- Divide by x² (highest denominator power): (3 + 1/x²)/(1 − 5/x²).
- Let x → infinity: 1/x² → 0 and 5/x² → 0.
- Survivors: 3/1 = 3. Horizontal asymptote y = 3.
- Degree shortcut check: equal degrees → ratio of leading coefficients: 3/1 = 3. Matches ✓
Your turn: limx→∞ (5x²−2)/(2x²+7) = ?
Answer: 5/2.
Divide by x²: (5 − 2/x²)/(2 + 7/x²) → 5/2. Horizontal asymptote y = 5/2.
Example 2 — denominator wins: lim(x→∞) (x+1)/(x²+3)
- Divide by x²: (1/x + 1/x²)/(1 + 3/x²).
- Let x → infinity: everything up top → 0; bottom → 1.
- Result: 0/1 = 0. Horizontal asymptote y = 0 (the x-axis).
Your turn: limx→∞ (4x−1)/(x³+2) = ?
Answer: 0.
Divide by x³: (4/x² − 1/x³)/(1 + 2/x³) → 0/1 = 0.
Example 3 — at minus infinity: lim(x→−∞) (2x³)/(x³−1)
- Divide by x³: 2/(1 − 1/x³).
- Let x → −infinity: 1/x³ → 0 (sign doesn’t matter for something → 0).
- Result: 2/1 = 2.
Your turn: limx→−∞ (3x³+1)/(x³−5) = ?
Answer: 3.
Divide by x³: (3 + 1/x³)/(1 − 5/x³) → 3/1 = 3.
Example 4 — the trap: lim(x→∞) (x²+1)/x
- Degree check: top degree 2 > bottom degree 1 → unbounded, no horizontal asymptote.
- Simplify to see how: (x²+1)/x = x + 1/x → +infinity (grows like x).
Your turn: limx→∞ (2x²+3)/x = ?
Answer: +∞.
Top degree wins — no horizontal asymptote. (2x²+3)/x = 2x + 3/x → +∞.
Memorization tips
- Divide by the highest denominator power: the one technique. Everything else on this page is commentary on it.
- The degree shortcut: top < bottom → 0; equal → ratio of leading coefficients; top > bottom → ±infinity. Three cases, five seconds.
- ∞/∞ is indeterminate: the x²/x, x/x², 2x/x trio (infinity, 0, 2) is the counterexample to memorize.
- N replaces delta: “eventually past N” is the at-infinity version of “within delta of a.” Same proof architecture.
- Asymptotes can be crossed: y = L describes the tail, not a wall. (sin x)/x crosses y = 0 infinitely often on its way there.
- Check −infinity separately when degrees differ — that’s where sign flips hide. Equal degrees: the ratio rules both ends.
Final challenge
Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and this law is yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
How do I compute a limit at infinity for a rational function?
Divide every term by the highest power of x in the denominator, then let x → ∞: all 1/xⁿ terms → 0, leaving the dominant balance. Shortcut: lower top degree → 0; equal degrees → ratio of leading coefficients; higher top degree → ±∞.
Why isn't ∞/∞ = 1?
It's indeterminate: lim(x→∞) x²/x = ∞, lim x/x² = 0, and lim 2x/x = 2. The ratio of growth rates decides — never a fixed rule.
What is a horizontal asymptote?
The line y = L where lim(x→∞) f(x) = L (or x → −∞). The graph settles toward it. The two ends may have different asymptotes, and the graph may cross it — it describes the tail, not a barrier.
What is the ε-N definition?
lim(x→∞) f(x) = L means: for every ε > 0 there is an N such that x > N implies |f(x) − L| < ε. 'Eventually, f stays within ε of L.' N plays δ's role at infinity.
Does x → −∞ ever give a different answer?
For equal degrees, no — the ratio of leading coefficients rules both ends. When the top degree is higher, signs can flip, so check −∞ separately.
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