MAT.126
1.2-1.3

1.2 Finding Limits Graphically and Numerically

Estimate a limit using a numerical or graphical approach

Numerical approach

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

Graphical approach

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Does the graph show a large irregularity of some sort?

Notation

Existence of f(x)

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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.

Learn different ways that a limit can fail to exist

Behavior that differs from the right and from the left

Unbounded behavior

Oscillating behavior

Dirichlet function

Study and use a formal definition of limit

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Epsilon-Delta definition of limit

1.3 Evaluating Limits Analytically

Evaluate a limit using properties of limits

Theorem 1.1
Some Basic Limits

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p. 59

Theorem 1.2
Properties of Limits

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p. 59

Theorem 1.3
Limits of Polynomials and Rational Functions

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p. 60

Theorem 1.4
The Limit of a Function Involving a Radical

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p. 60

Theorem 1.5
The Limit of a Composite Function

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p. 61

Theorem 1.6
Limits of Trigonometric Functions

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p. 61

Develop and use a strategy for finding limits

Theorem 1.7
Functions That Agree At All But One Point

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p.62

A Strategy for Finding Limits

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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.

Evaluate a limit using dividing out and rationalizing techniques

Dividing Out

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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.

Rationalizing

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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.

Evaluate a limit using the Squeeze Theorem

Theorem 1.8
The Squeeze Theorem

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p. 65

Theorem 1.9
Two Special Trigonometric Limits

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p. 65