One-Sided Limits

Key idea

Left and right limits must agree for the limit to exist.

In depth

The left limit uses inputs less than a, and the right limit uses inputs greater than a. The two-sided limit exists only if both one-sided limits exist and are equal. They are essential for jumps and endpoints.

Worked example

For f(x) = |x|/x, the limit as x → 0 from the right is 1 and from the left is −1, so the two-sided limit does not exist.

Real-world uses

• Modelling tax or postage bands where the price jumps at a boundary
• The flow rate at a valve opening

Common mistake

Assuming a limit exists because one side does.

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.
• 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.
• Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.
• Squeeze Theorem – If g ≤ f ≤ h and g, h → L then f → L.