Lesson

What a limit is allowed to ignore

The value at a single x does not get to veto the approaching height.

Learning goals

Estimate a limit from a graph or a table.
Explain why an undefined point can still have a limit.

Explanation

The sentence “the limit as x approaches 2 is 5” means that f(x) gets and stays arbitrarily close to 5 when x is close to 2, from both sides if we mean a two-sided limit. What happens exactly at 2 can be a hole, a different y-value, or a missing input.

Tables can lie if you only look at one side or if you stop too far from the point. Graphs can hide a hole behind a printed dot. Algebra can cancel a common factor and reveal the height the hole was hiding. Use more than one view when they disagree.

Key terms

Limit. The value a function approaches as the input approaches a number, if that approaching value exists.
Two-sided limit. A limit that requires the left-hand and right-hand approaching values to match.

Common mistakes

Reporting f(a) when asked for a limit, or the reverse.
Declaring a limit does not exist because a table never shows the exact target number.

Practice

Original Marlow Works items. Check the answer explanation after you try.

1 of 1Limits and Continuity · easy · multiple choice

What is lim (x→3) of (x² − 9) / (x − 3)?

Original Marlow Works item — not a College Board question.

Take your time—this is practice, not a test.

Answer explanation

A graph has a hole at x = 1 sitting at y = 4, and a filled dot at (1, 2). What is lim x→1 f(x)?

Answer. 4, if both sides approach the hole’s height.

The limit cares about nearby points, not the filled-in value.

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Related resources

External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.

Reference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source