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Infinite Limit

Infinite Limit

A limit that has an infinite result (either ∞ or –∞ ), or a limit taken as the variable approaches ∞ (infinity) or –∞ (minus infinity). The limit can be one-sided.

 

 

See also

Limit from the left, limit from the right

Key Formula

lim⁡x→cf(x)=±∞orlim⁡x→±∞f(x)=L\lim_{x \to c} f(x) = \pm\infty \qquad \text{or} \qquad \lim_{x \to \pm\infty} f(x) = L
Where:
  • xx = The independent variable approaching a value or infinity
  • cc = A finite value that x approaches
  • f(x)f(x) = The function being evaluated
  • LL = The resulting limit value (which may itself be finite or infinite)

Worked Example

Problem: Evaluate the limit: lim⁡x→01x2\lim_{x \to 0} \dfrac{1}{x^2}.
Step 1: Consider what happens as xx gets closer and closer to 00 from either side. Try values like x=0.1x = 0.1, x=0.01x = 0.01, and x=0.001x = 0.001.
f(0.1)=10.01=100,f(0.01)=10.0001=10,000,f(0.001)=100,000,000f(0.1) = \frac{1}{0.01} = 100, \quad f(0.01) = \frac{1}{0.0001} = 10{,}000, \quad f(0.001) = 100{,}000{,}000
Step 2: Since x2x^2 is always positive (whether xx approaches 00 from the left or the right), the fraction 1x2\frac{1}{x^2} is always positive and grows without bound.
x2>0 for all x≠0x^2 > 0 \text{ for all } x \neq 0
Step 3: Because the function increases without bound as x→0x \to 0, the limit is positive infinity.
lim⁡x→01x2=+∞\lim_{x \to 0} \frac{1}{x^2} = +\infty
Answer: The limit is +∞+\infty. The function grows without bound as xx approaches 00.

Another Example

Problem: Evaluate the limit: lim⁡x→0+1x\lim_{x \to 0^+} \dfrac{1}{x}.
Step 1: This is a one-sided limit from the right. Consider values of xx that are small and positive: x=0.1x = 0.1, x=0.01x = 0.01, x=0.001x = 0.001.
f(0.1)=10,f(0.01)=100,f(0.001)=1000f(0.1) = 10, \quad f(0.01) = 100, \quad f(0.001) = 1000
Step 2: As xx approaches 00 from the positive side, 1x\frac{1}{x} increases without bound.
lim⁡x→0+1x=+∞\lim_{x \to 0^+} \frac{1}{x} = +\infty
Step 3: Note that from the left (x→0−x \to 0^-), the values are negative and decrease without bound, giving −∞-\infty. Because the two one-sided limits differ, lim⁡x→01x\lim_{x \to 0} \frac{1}{x} does not exist.
lim⁡x→0−1x=−∞\lim_{x \to 0^-} \frac{1}{x} = -\infty
Answer: The limit from the right is +∞+\infty, the limit from the left is −∞-\infty, so the two-sided limit does not exist.

Frequently Asked Questions

Does an infinite limit mean the limit exists?
Technically, when we write lim⁡x→cf(x)=∞\lim_{x \to c} f(x) = \infty, the limit does not exist in the usual finite sense. Saying the limit "equals infinity" is a shorthand that describes the function's behavior — it grows without bound. In formal mathematics, a limit exists only when it equals a finite real number.
What is the difference between a limit at infinity and an infinite limit?
A limit at infinity evaluates what happens to f(x)f(x) as xx itself grows toward ±∞\pm\infty (e.g., lim⁡x→∞1x=0\lim_{x \to \infty} \frac{1}{x} = 0). An infinite limit describes the function's output blowing up to ±∞\pm\infty as xx approaches some finite value (e.g., lim⁡x→01x2=+∞\lim_{x \to 0} \frac{1}{x^2} = +\infty). The term "infinite limit" is sometimes used broadly to cover both situations.

Infinite Limit (output → ±∞) vs. Limit at Infinity (input → ±∞)

An infinite limit means the function's value becomes arbitrarily large (or large in the negative direction) as xx approaches a finite number. A limit at infinity asks what value the function approaches as xx itself heads toward +∞+\infty or −∞-\infty. For example, lim⁡x→01x2=+∞\lim_{x \to 0} \frac{1}{x^2} = +\infty is an infinite limit, while lim⁡x→∞1x=0\lim_{x \to \infty} \frac{1}{x} = 0 is a limit at infinity with a finite result.

Why It Matters

Infinite limits reveal where a function has vertical asymptotes — places on the graph where the curve shoots upward or downward without bound. Recognizing these behaviors is essential for sketching accurate graphs and understanding function behavior in calculus. They also appear in physics and engineering when modeling quantities that blow up near critical points, such as electric field strength near a point charge.

Common Mistakes

Mistake: Writing lim⁡x→01x=∞\lim_{x \to 0} \frac{1}{x} = \infty without checking both sides.
Correction: For 1x\frac{1}{x}, the limit from the right is +∞+\infty and from the left is −∞-\infty. Since the one-sided limits disagree, the two-sided limit does not exist. Always check both sides when the function can change sign near the point.
Mistake: Treating ∞\infty as a real number and performing arithmetic with it (e.g., ∞−∞=0\infty - \infty = 0).
Correction: Infinity is not a number. Expressions like ∞−∞\infty - \infty are indeterminate forms that require further analysis (such as algebraic simplification or L'Hôpital's Rule) to evaluate.

Related Terms