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1.3 Evaluating Limits Analytically

Evaluate a limit using the Squeeze Theorem

Theorem 1.9 Two Special Trigonometric Limits

Theorem 1.8 The Squeeze Theorem

p. 65

Evaluate a limit using dividing out and rationalizing techniques

Rationalizing

Use when rational limits with radicals have the form 0/0 after direct substitution.

The expression 0/0 is indeterminate.

Theorem 1.7 allows this method to work.

Dividing Out

Use when rational limits have the form 0/0 after direct substitution.

The expression 0/0 is indeterminate.

Think "simplify."

Theorem 1.7 allows this method to work.

Develop and use a strategy for finding limits

A Strategy for Finding Limits

Learn to recognize which limits can be evaluated by direct substitution (Theorems 1.1-1.6).

If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x=c. [Choose g such that the limit of g(x) can be evaluated by direct substitution.]

Apply Theorem 1.7 to conclude analytically that your choice of g works.

Use a graph or table to reinforce your conclusion.

Theorem 1.7 Functions That Agree At All But One Point

p.62

Evaluate a limit using properties of limits

Theorem 1.6 Limits of Trigonometric Functions

Theorem 1.5 The Limit of a Composite Function

p. 61

Theorem 1.4 The Limit of a Function Involving a Radical

Theorem 1.3 Limits of Polynomials and Rational Functions

p. 60

Theorem 1.2 Properties of Limits

Theorem 1.1 Some Basic Limits

p. 59

1.2 Finding Limits Graphically and Numerically

Study and use a formal definition of limit

Epsilon-Delta definition of limit

Learn different ways that a limit can fail to exist

Oscillating behavior

Dirichlet function

Unbounded behavior

Behavior that differs from the right and from the left

Estimate a limit using a numerical or graphical approach

Existence of f(x)

The existence or nonexistence of f(x) at x=c has no bearing on the existence of the limit of f(x) as x approaches c.

Notation

Graphical approach

Does the graph show a large irregularity of some sort?

Numerical approach

Use a table of values that approach the value of interest from both sides.