1.6 Determining Limits Using Algebraic Manipulation

Learning objectives

—LIM-1.E—Determine the limits of functions using equivalent expressions for the function or the squeeze theorem• LIM-1.E Determine the limits of functions using equivalent expressions for the function or the squeeze theorem.• LIM-1.E.1 It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.• Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

If direct substitution gives $0/0$ , the result is indeterminate—not the value of the limit. Rewrite the expression into an equivalent form that is valid for inputs near the target, then evaluate the simpler limit.

Structure causing $0/0$	Useful rewrite
Numerator and denominator share a polynomial factor	Factor, then divide out the common factor
A difference involving square roots	Multiply by the appropriate conjugate
A trigonometric expression in an unhelpful form	Use an identity to create a recognizable equivalent form

\begin{aligned}\lim_{x\to3}\frac{x^2-9}{x-3}&=\lim_{x\to3}\frac{(x-3)(x+3)}{x-3}\&=\lim_{x\to3}(x+3)\qquad(x\ne3)\&=6.\end{aligned}

The original quotient is undefined at $x=3$ , but for every nearby input with $x\ne3$ it equals $x+3$ . Because a limit uses nearby behavior rather than the value at the point, both expressions have the same limit as $x$ approaches $3$ .

Cancel only common factors, not separate terms in a sum or difference. After rewriting, check that the new expression is genuinely equal to the original for all sufficiently close inputs except possibly the target itself. The squeeze theorem is a separate method developed in Topic 1.8.

Next Topic

—1.7 Selecting Procedures for Determining LimitsRead