Calculus I › Limits › full formula sheet
Infinite limits
When the function grows without bound near a point — and why the sign decides everything.
On this page “= ∞” is shorthand, not an equation: it describes how the limit fails to exist (by growing without bound). The two-sided infinite limit exists only if both sides agree in sign.
Notation in this lesson
- limx→a f(x) = ±∞
- f grows without bound near a — the values explode
- x = a
- vertical asymptote — the line the graph hugs
- M
- any large bound: f(x) exceeds every M near a
- δ
- how close x must stay to a to force f(x) > M
Where it comes from
Infinite limits are born where the quotient law dies: numerator heading somewhere nonzero, denominator heading to zero. The function doesn’t approach a number — it grows without bound, and we write lim = ±infinity as precise shorthand for that behavior.
The naive move is “1/0 = infinity” — wrong twice:
The intuition: a nonzero numerator over a vanishing denominator means the ratio’s magnitude explodes; the signs of numerator and denominator decide whether it explodes upward or downward. The derivation makes “grows without bound” rigorous with M playing epsilon’s role.
Derivation
The definition mirrors epsilon-delta, with a bound M replacing epsilon. Then we prove lim(x→0) 1/x² = +infinity as the model argument.
Key steps shown; the argument above is complete. For −infinity, negate: lim f = −infinity means lim(−f) = +infinity, i.e. f(x) < −M near a for every M > 0. And “the limit is infinity” never means the limit exists — it’s the precise description of one way a limit can fail.
How to use it
The procedure, every time:
- Identify the shape: numerator → nonzero L, denominator → 0. That’s the infinite-limit signature (the quotient law is off).
- Go one-sided. Determine the sign from each side: sign(numerator) × sign(denominator). Even powers in the denominator give the same sign both sides; odd powers flip.
- Compare sides: both +infinity (or both −infinity) → two-sided infinite limit. Mixed → the two-sided limit DNE.
- Conclude the asymptote: x = a is a vertical asymptote if at least one one-sided limit is infinite.
Reading the sign fast
Near a, ask: is the numerator positive or negative? Is the denominator approaching 0 through positive or negative values? +/+ and −/− explode upward; +/− and −/+ explode downward. For (x+2)/(x−1)³ at x = 1: numerator → 3 > 0; denominator → 0+ from the right (+infinity), 0− from the left (−infinity).
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — both sides agree: lim(x→0) 1/x²
- Signature check: numerator → 1 (nonzero), denominator → 0. Infinite behavior.
- Signs: x² > 0 from both sides (even power) — 1/0+ on both sides.
- Conclude: +infinity (two-sided). Vertical asymptote x = 0.
Your turn: limx→0 2/x² = ?
Answer: +∞.
Numerator → 2 (nonzero), denominator → 0+ from both sides (x² > 0). 2/0+ = +∞; vertical asymptote x = 0.
Example 2 — still agreeing: lim(x→2) 3/(x−2)²
- Signature: numerator → 3, denominator → 0.
- Signs: (x−2)² > 0 both sides; numerator positive. Both sides: 3/0+.
- Conclude: +infinity. Vertical asymptote x = 2.
Your turn: limx→1 4/(x−1)² = ?
Answer: +∞.
Denominator → 0+ on both sides; numerator → 4. Both sides agree: +∞.
Example 3 — sides disagree: lim(x→0) 1/x
- Right side: x → 0+: 1/0+ = +infinity.
- Left side: x → 0−: 1/0− = −infinity.
- Two-sided verdict: the sides disagree, so the two-sided limit DNE — even as an infinite limit. (x = 0 is still a vertical asymptote: one infinite side suffices.)
Your turn: limx→0 1/x³ = ?
Answer: DNE (two-sided).
Right side: 1/0+ = +∞; left side: 1/0− = −∞. The sides disagree, so no two-sided limit — even as an infinite limit.
Example 4 — odd power flips: lim(x→1) (x+2)/(x−1)³
- Signature: numerator → 3 (nonzero), denominator → 0.
- Right side: (x−1)³ → 0+ (positive small cubed stays positive): 3/0+ = +infinity.
- Left side: (x−1)³ → 0−: 3/0− = −infinity.
- Verdict: two-sided limit DNE; vertical asymptote x = 1.
Your turn: limx→2 (x+1)/(x−2)³ = ?
Answer: DNE (two-sided).
Numerator → 3; right side 3/0+ = +∞, left side 3/0− = −∞. Vertical asymptote x = 2.
Memorization tips
- Numerator nonzero + denominator zero = blowup: the signature. The quotient law bows out; one-sided analysis takes over.
- Even powers agree, odd powers flip: (x−a)2n gives the same infinity both sides; (x−a)2n+1 gives opposite infinities.
- Sign(numerator) × sign(denominator): the two-sign multiplication that decides +infinity vs −infinity. Do it explicitly every time.
- “DNE” is still the honest two-sided verdict when the sides disagree — “infinity” describes how it fails, one side at a time.
- One infinite side makes the asymptote: x = a is a vertical asymptote if either one-sided limit is infinite.
- M replaces epsilon: the definition “for every bound M, f exceeds M near a” is the infinite analogue of epsilon-delta. Same architecture, unbounded conclusion.
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
What does lim(x→a) f(x) = ∞ actually mean?
It's shorthand, not an equation: for every bound M > 0, f(x) exceeds M near a. It describes precisely HOW the limit fails to exist — by growing without bound.
Why is '1/0 = ∞' wrong?
1/0 is meaningless, and the sign depends on the side: lim(x→0⁺) 1/x = +∞ but lim(x→0⁻) 1/x = −∞. Always do a one-sided sign analysis instead.
When does the two-sided infinite limit exist?
Only when both one-sided limits are infinite with the SAME sign — e.g. lim(x→0) 1/x² = +∞. For 1/x the sides disagree, so the two-sided limit DNE.
How do I find the sign quickly?
Multiply sign(numerator) × sign(denominator) near a from each side. Even powers in the denominator give the same sign both sides; odd powers flip it.
Does an infinite limit mean there's a vertical asymptote?
Yes — x = a is a vertical asymptote if at least one one-sided limit is ±∞. Both sides need not agree.
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