Topic guide · Calculus

Limits and derivatives: where to begin

Calculus becomes easier when you connect the algebra to its underlying idea: how a quantity behaves as an input approaches a value.

What is a limit?

A limit describes the value a function approaches as x gets close to a point. The function does not need to be defined at that point for a limit to exist. To establish a two-sided limit, both the left-hand and right-hand limits must exist and agree.

Worked example: simplify first

Consider the limit of (x² − 1)/(x − 1) as x approaches 1. Direct substitution gives 0/0, which is indeterminate. For x ≠ 1, factor the numerator to get (x − 1)(x + 1), so the expression equals x + 1. As x approaches 1, this approaches 2. We have found the limit, not assigned a value to the original expression at x = 1.

The derivative as a rate of change

The derivative of f at x is the limit, if it exists, of [f(x + h) − f(x)]/h as h approaches 0. Geometrically it gives the slope of the tangent to the graph at that point; in an application it can represent an instantaneous rate of change.

Worked example: derivative of x²

Let f(x) = x². Then [f(x + h) − f(x)]/h = [(x + h)² − x²]/h = 2x + h for h ≠ 0. As h approaches zero, this tends to 2x. Therefore the derivative is f′(x) = 2x, and the slope at x = 3 is 6.

Common pitfalls and practice

• Do not treat 0/0 as an answer: simplify or use an appropriate method.
• Check both sides when a function is piecewise-defined or has an absolute value.
• Keep track of the point at which a derivative is evaluated.

Try: What is the limit of (x² − 4)/(x − 2) as x approaches 2? What is the slope of y = x² at x = −1?

Answers: Factoring gives x + 2 for x ≠ 2, so the limit is 4. The slope is 2(−1) = −2.