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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
Limits of Exponentials at Infinity
Limits of Logarithms Toward Zero and Infinity
Summary: Master Diagnostic for Limits and Asymptotes



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.

    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 $m < n$: 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 $m = n$: 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 $m > n$: 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:

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


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

Limits of Exponentials at Infinity


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

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


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


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.

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