Infinite Limits and Asymptotes

Key idea

Function values grow without bound near a vertical asymptote.

In depth

If f(x) grows without bound near x = a, the limit is ±∞ and the line x = a is a vertical asymptote. It usually happens where the denominator is zero and the numerator is not. Check the sign from each side.

Worked example

lim 1/(x − 1)² as x → 1 is +∞; lim 1/(x − 1) is +∞ from the right and −∞ from the left.

Formula

Real-world uses

• Modelling how average cost blows up at zero output or how signal strength diverges

Common mistake

Saying the limit equals ∞ as if it were a real number; it means unbounded growth.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Calculus

• Idea of a Limit – The value a function approaches as the input approaches a point.
• One-Sided Limits – Left and right limits must agree for the limit to exist.
• Limit Laws – Limits of sums, products and quotients are computed from the parts.
• Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
• Limits at Infinity – End behaviour of functions.
• Squeeze Theorem – If g ≤ f ≤ h and g, h → L then f → L.