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Limits

Distinguish a function’s value at a point from the value it approaches nearby. Work through a removable gap and explain why direct substitution can be misleading.

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What you’ll learn

Explain what a limit describes.
Distinguish a limit from a function’s value at the point.
Use a valid simplification to investigate a limit.

A limit concerns nearby inputs

A limit describes the value approached as the input gets close to a point. The function may have that value at the point, a different value, or no value there at all.

Use the expression and the graph together

A graph or table can suggest behavior, but the expression and its conditions help justify it. When simplifying a quotient, keep track of the values where the simplification is valid.

A mistake to avoid

The expression 0/0 does not give a numerical answer. It signals that direct substitution has not resolved the limit. Also compare behavior from both sides when evaluating a two-sided limit.

Worked example

Approach x = 1 in (x² − 1)/(x − 1)

For x ≠ 1, factor the numerator to get (x − 1)(x + 1)/(x − 1) = x + 1. As x approaches 1, this approaches 2. The original expression remains undefined at x = 1; the limit is 2 nonetheless.

Learning paths that include this skill

Choose the route that fits your goal. The same skill can be useful in more than one subject or stage of learning.

Preparing for calculusPrecalculus foundationsReview equation, graph, and angle foundations before calculus. Identify the prerequisite that needs attention and connect formal notation to its meaning.Linear equations · Systems of equations · Quadratic equations · Graphs & coordinates · +5 more Introductory university calculusUniversity calculus foundationsConnect the first calculus ideas to functions, graphs, and justified steps. Understand local change and accumulation before extending the techniques.Graphs & coordinates · Function notation · Limits · Derivatives · +2 more

DerivativesConnect a derivative to a local rate of change. Use a difference quotient to explain the tangent slope and distinguish it from the function’s height.IntegralsInterpret a definite integral as accumulated change and connect it to an antiderivative. Keep bounds, signs, and units distinct from an indefinite integral.

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