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Infinite Limits and Limits at Infinity



Two Distinct Concepts
Limits at Infinity — Horizontal Behavior
Horizontal Asymptotes
Evaluating Limits at Infinity — Rational Functions
Dominant Term Analysis
Growth Rates — The Hierarchy
Infinite Limits — Vertical Behavior
Vertical Asymptotes
Sign Analysis for Infinite Limits
One-Sided Infinite Limits
Infinity Symbol Notation
Limits of Exponentials at Infinity
Limits of Logarithms Toward Zero and Infinity
Summary: Master Diagnostic for Limits and Asymptotes
Limits and Infinity FAQ



Beyond All Bounds


Infinity enters limits in two distinct ways. When we write x→∞x \to \infty, we ask what happens to a function as its input grows without bound. When we write f(x)→∞f(x) \to \infty, we describe a function whose output grows without bound as the input approaches some value.

These are different phenomena requiring different analyses. Limits at infinity reveal end behavior—how a function settles (or doesn't) as xx moves far from the origin. Infinite limits reveal explosive behavior—where a function blows up as xx approaches a finite point.

Both concepts connect to asymptotes. Horizontal asymptotes arise from finite limits at infinity. Vertical asymptotes arise from infinite limits at finite points. Together they sketch a function's large-scale structure.

Key Terms

Asymptote— horizontal (y=Ly = L) and vertical (x=ax = a) boundary lines
Limit— limits at infinity and infinite limits
One-Sided Limit— one-sided infinite limits at vertical asymptotes

See All Calculus Definitions →


Limits at Infinity — Horizontal Behavior


The limit

lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L


means f(x)f(x) approaches LL as xx increases without bound. No matter how close to LL you demand, sufficiently large xx will place f(x)f(x) within that tolerance.

Similarly:

lim⁡x→−∞f(x)=M\lim_{x \to -\infty} f(x) = M


means f(x)f(x) approaches MM as xx decreases without bound.

These limits describe end behavior—the function's long-run tendency. The limit may be a finite number, ∞\infty, −∞-\infty, or may not exist (if the function oscillates).

Horizontal Asymptotes


    If lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L where LL is finite, the line y=Ly = L is a horizontal asymptote. The graph approaches this line as x→∞x \to \infty.

    Horizontal Asymptote Condition
    lim⁡x→∞f(x)=L   or   lim⁡x→−∞f(x)=L  ⟹  y=L is a horizontal asymptote\lim_{x \to \infty} f(x) = L \;\text{ or }\; \lim_{x \to -\infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}
    Learn more about this formula: Horizontal Asymptote Condition →


    A function can have:

  • x→∞x \to \infty and x→−∞x \to -\infty)

    • For example:

    f(x)=2x+1x−3f(x) = \frac{2x + 1}{x - 3}


    lim⁡x→∞f(x)=2lim⁡x→−∞f(x)=2\lim_{x \to \infty} f(x) = 2 \qquad \lim_{x \to -\infty} f(x) = 2


    The line y=2y = 2 is a horizontal asymptote in both directions.

Evaluating Limits at Infinity — Rational Functions


For rational functions, divide numerator and denominator by the highest power of xx in the denominator.

lim⁡x→∞3x2+5x−12x2−7\lim_{x \to \infty} \frac{3x^2 + 5x - 1}{2x^2 - 7}


Divide by x2x^2:

=lim⁡x→∞3+5x−1x22−7x2=3+0−02−0=32= \lim_{x \to \infty} \frac{3 + \frac{5}{x} - \frac{1}{x^2}}{2 - \frac{7}{x^2}} = \frac{3 + 0 - 0}{2 - 0} = \frac{3}{2}


Terms with xx in the denominator vanish as x→∞x \to \infty. Only the leading coefficients survive.

Dominant Term Analysis


As x→∞x \to \infty, the highest-degree terms control behavior. Lower-degree terms become negligible.

For p(x)q(x)\dfrac{p(x)}{q(x)} where pp has degree mm and qq has degree nn:

Case m<nm < n: The limit is 00. The denominator grows faster.

lim⁡x→∞x+1x3−2=0\lim_{x \to \infty} \frac{x + 1}{x^3 - 2} = 0


Case m=nm = n: The limit is the ratio of leading coefficients.

lim⁡x→∞4x2+x2x2+5=42=2\lim_{x \to \infty} \frac{4x^2 + x}{2x^2 + 5} = \frac{4}{2} = 2


Case m>nm > n: The limit is ±∞\pm\infty. The numerator grows faster.

lim⁡x→∞x3x+1=∞\lim_{x \to \infty} \frac{x^3}{x + 1} = \infty

Degree relation limx → ∞ p(x)/q(x) Why Example
deg p < deg q 0 denominator grows faster than numerator (x + 1) / (x³ − 2) → 0
deg p = deg q ratio of leading coefficients scales tie; only leading terms survive (4x² + x) / (2x² + 5) → 4 / 2 = 2
deg p > deg q ±∞ numerator grows faster than denominator; sign from leading-coefficient ratio x³ / (x + 1) → ∞

Growth Rates — The Hierarchy


Different function types grow at fundamentally different rates as x→∞x \to \infty:

logarithmic≪polynomial≪exponential\text{logarithmic} \ll \text{polynomial} \ll \text{exponential}


Any exponential eventually overtakes any polynomial:

lim⁡x→∞xnex=0for any n\lim_{x \to \infty} \frac{x^n}{e^x} = 0 \quad \text{for any } n


Any polynomial eventually overtakes any logarithm:

lim⁡x→∞ln⁡xxn=0for any n>0\lim_{x \to \infty} \frac{\ln x}{x^n} = 0 \quad \text{for any } n > 0


These special limits determine which terms dominate in mixed expressions.

Infinite Limits — Vertical Behavior


The notation

lim⁡x→af(x)=∞\lim_{x \to a} f(x) = \infty


means f(x)f(x) grows without bound as xx approaches aa. For any large number MM, values of xx sufficiently close to aa make f(x)>Mf(x) > M.

Similarly:

lim⁡x→af(x)=−∞\lim_{x \to a} f(x) = -\infty


means f(x)f(x) becomes arbitrarily negative.

These are not limits in the usual sense—∞\infty is not a number. The notation describes unbounded behavior, not convergence to a value.

Vertical Asymptotes


If any of the following hold:

Vertical Asymptote Condition
lim⁡x→a−f(x)=±∞   or   lim⁡x→a+f(x)=±∞  ⟹  x=a is a vertical asymptote\lim_{x \to a^-} f(x) = \pm\infty \;\text{ or }\; \lim_{x \to a^+} f(x) = \pm\infty \implies x = a \text{ is a vertical asymptote}
Learn more about this formula: Vertical Asymptote Condition →


then the line x=ax = a is a vertical asymptote. The graph shoots up or down near x=ax = a.

Vertical asymptotes typically occur where the denominator of a rational function equals zero while the numerator does not.

For f(x)=1x−2f(x) = \dfrac{1}{x - 2}, the denominator vanishes at x=2x = 2. The line x=2x = 2 is a vertical asymptote.

Sign Analysis for Infinite Limits


To determine whether a limit is +∞+\infty or −∞-\infty, analyze the sign of the expression near the point.

lim⁡x→2+1x−2\lim_{x \to 2^+} \frac{1}{x - 2}


For xx slightly greater than 22: x−2>0x - 2 > 0 (small positive), so 1x−2\dfrac{1}{x-2} is large positive.

lim⁡x→2+1x−2=+∞\lim_{x \to 2^+} \frac{1}{x - 2} = +\infty


For xx slightly less than 22: x−2<0x - 2 < 0 (small negative), so 1x−2\dfrac{1}{x-2} is large negative.

lim⁡x→2−1x−2=−∞\lim_{x \to 2^-} \frac{1}{x - 2} = -\infty


One-Sided Infinite Limits


The one-sided limits from the left and right may both be infinite but with opposite signs.

For f(x)=1x−2f(x) = \dfrac{1}{x - 2}:

lim⁡x→2−f(x)=−∞lim⁡x→2+f(x)=+∞\lim_{x \to 2^-} f(x) = -\infty \qquad \lim_{x \to 2^+} f(x) = +\infty


The two-sided limit does not exist because the sides disagree—even though both are "infinite."

For f(x)=1(x−2)2f(x) = \dfrac{1}{(x-2)^2}:

lim⁡x→2−f(x)=+∞lim⁡x→2+f(x)=+∞\lim_{x \to 2^-} f(x) = +\infty \qquad \lim_{x \to 2^+} f(x) = +\infty


Both sides agree, so we write lim⁡x→2f(x)=+∞\lim_{x \to 2} f(x) = +\infty.
Function near x = a LHL at a RHL at a Two-sided Asymptote pattern
1 / (x − a)  (odd power) −∞ +∞ does not exist (signs disagree) vertical asymptote; graph plunges on one side, soars on the other
1 / (x − a)²  (even power) +∞ +∞ +∞ (sides agree) vertical asymptote; graph soars on both sides

Infinity Symbol Notation

Notation

Infinity Symbol Notation

One glyph, opposite jobs: under the lim as a direction for xx, after the equals sign as a verdict about ff. The entries keep them apart.
lim⁡\lim, the bare arrow and DNE belong to two-sided limits; the superscripts a−a^{-}, a+a^{+} to one-sided limits.
∞\infty
Infinity
Not a number — a name for unboundedness. It never enters arithmetic and appears only inside limit statements. LaTeX \infty; the mathematical keyboard carries ∞\infty.
Cases+∞+\infty and −∞-\infty — the two ends of the real line. Bare ∞\infty conventionally means +∞+\infty in real calculus.
Also written+∞+\infty spelled with its sign in French and Eastern European texts, which reserve the bare glyph for the unsigned point of projective geometry.
Do not confuseA number that obeys arithmetic. ∞−∞\infty - \infty and ∞/∞\infty/\infty have no fixed value — they are the indeterminate forms of limit evaluation.
Same glyph elsewhereIn set theory infinite sizes are ℵ₀, 𝔠 — provably different infinities, a distinction the calculus symbol never makes.
x→∞x \to \infty
As x approaches infinity
The input escapes: eventually beyond every bound. No point is being approached — shorthand, not location.
Casesx→−∞x \to -\infty — the other end; the two directions are independent, which is how a function earns two different asymptotes in Horizontal Asymptotes above. With an integer index, n→∞n \to \infty, the same notation runs through sequences and series.
Also writtenx→+∞x \to +\infty, matching the signed tradition above.
Do not confuse“Plug in a huge number.” Nothing is substituted — ∞\infty never enters ff; the limit describes the trend, in Limits at Infinity — Horizontal Behavior above.
lim⁡x→af(x)=∞\lim\limits_{x \to a} f(x) = \infty
The limit is infinity
An abuse of the equals sign that stuck. The limit does not exist — but fails in the most informative way, unbounded growth, so the == records behavior rather than a value.
CasesOne-sided versions carry the superscripts and diagnose the lines in Vertical Asymptotes above, sign determined as in Sign Analysis above. The two-sided form is only legal when both sides agree on the same signed infinity — One-Sided Infinite Limits above.
Also writtenf(x)→∞f(x) \to \infty as x→ax \to a — arrow form. Stricter analysis texts refuse the == entirely and write “increases without bound”.
Do not confuse“Exists, and its value is infinity.” In real-valued calculus it does not exist; only on the extended real line is the equation literal — which is why the abuse survives.

Limits of Exponentials at Infinity


The exponential function exe^x exhibits contrasting behavior in opposite directions:

Exponential End Behavior
lim⁡x→∞ex=∞,lim⁡x→−∞ex=0\lim_{x \to \infty} e^x = \infty, \qquad \lim_{x \to -\infty} e^x = 0
Learn more about this formula: Exponential End Behavior →


As x→∞x \to \infty, exponential growth is unbounded. As x→−∞x \to -\infty, the function decays toward zero.

For e−xe^{-x}, the behavior reverses:

lim⁡x→∞e−x=0\lim_{x \to \infty} e^{-x} = 0


lim⁡x→−∞e−x=∞\lim_{x \to -\infty} e^{-x} = \infty


The horizontal asymptote y=0y = 0 appears in the direction where the exponent goes to −∞-\infty.

Limits of Logarithms Toward Zero and Infinity


The natural logarithm is defined only for x>0x > 0. Its behavior at the boundaries:

lim⁡x→∞ln⁡x=∞\lim_{x \to \infty} \ln x = \infty


The logarithm grows without bound, but slowly—slower than any positive power of xx.

lim⁡x→0+ln⁡x=−∞\lim_{x \to 0^+} \ln x = -\infty


As xx approaches zero from the right, ln⁡x\ln x plunges to −∞-\infty. The line x=0x = 0 is a vertical asymptote for ln⁡x\ln x.

Logarithm End Behavior
lim⁡x→∞ln⁡x=∞,lim⁡x→0+ln⁡x=−∞\lim_{x \to \infty} \ln x = \infty, \qquad \lim_{x \to 0^+} \ln x = -\infty
Learn more about this formula: Logarithm End Behavior →


The one-sided notation is essential: ln⁡x\ln x is undefined for x≤0x \leq 0, so no left-hand limit exists at x=0x = 0.
Function Direction Limit Asymptote signal
ex x → ∞ ∞ no horizontal asymptote on the right
ex x → −∞ 0 horizontal asymptote y = 0 on the left
e−x x → ∞ 0 horizontal asymptote y = 0 on the right
e−x x → −∞ ∞ no horizontal asymptote on the left
ln x x → ∞ ∞ no horizontal asymptote; grows slowly (slower than any positive power)
ln x x → 0⁺ −∞ vertical asymptote x = 0 (right-hand limit only — ln undefined for x ≤ 0)

Summary: Master Diagnostic for Limits and Asymptotes


Every type of infinity behavior on this page maps cleanly to a graph feature: finite limits at ±∞ create horizontal asymptotes, infinite limits at finite a create vertical asymptotes, and the relative degree of a rational function&apos;s numerator and denominator determines exactly which case applies. The table below collects the diagnostic patterns in one master reference — given an observed limit form, it points to what that form means and what asymptote (if any) it produces.
Observed pattern What it signals Resulting graph feature Section
limx → ±∞ f(x) = L  (finite) end behavior settles to a single value horizontal asymptote y = L in that direction obj3
limx → ±∞ f(x) = ±∞ function grows without bound in that direction no horizontal asymptote there obj5
limx → a f(x) = ±∞  (one or both sides) output blows up near a finite point a vertical asymptote x = a obj8, obj9
Rational p/q,  deg p < deg q denominator dominates horizontal asymptote y = 0 obj4, obj5
Rational p/q,  deg p = deg q leading-degree terms balance horizontal asymptote y = (leading p) / (leading q) obj4, obj5
Rational p/q,  deg p > deg q numerator dominates no horizontal asymptote (function escapes to ±∞) obj5
Mixed exponential / logarithmic apply the growth-rate hierarchy (log ≪ poly ≪ exp) depends on which family dominates — see obj11, obj12 above obj6, obj11, obj12

Limits and Infinity FAQ

What is the difference between limits at infinity and infinite limits?

+
They describe opposite situations. A limit at infinity asks what f(x) does as x escapes beyond every bound; a finite answer gives a horizontal asymptote. An infinite limit asks what happens as x approaches a finite point where f grows without bound, which gives a vertical asymptote. The first puts ∞ under the lim, the second after the equals sign.Read more →

Is infinity a number?

+
No. In calculus ∞ is a name for unboundedness rather than a value, and it never enters arithmetic — which is why ∞ − ∞ and ∞/∞ are indeterminate forms instead of computations. It appears only inside limit statements. Set theory does treat infinite sizes as objects and distinguishes ℵ₀ from 𝔠, but the calculus symbol draws no such distinction.Read more →

Does x → ∞ mean plugging in a huge number?

+
No. Nothing is substituted — ∞ never enters the function. The notation says the input escapes beyond every bound, and the limit records the resulting trend in f. Testing one large value can suggest an answer but never establishes it, because the behavior has to hold past every bound, not past a single chosen number.Read more →

What is the limit of e^x as x approaches negative infinity?

+
It is 0. As x decreases without bound, e^x decays toward zero without ever reaching it, making the x-axis a horizontal asymptote on the left. The other direction reverses completely: as x increases without bound, e^x grows without bound, so there is no horizontal asymptote on the right.Read more →