1.3 Evaluating Limits Analytically

Objective: Evaluate a limit using properties of limits

Miss BattagliaAB/BC Calculus

Properties of Limits

• Remember that the limit of f(x) as x approaches c does not depend on the value of f at x=c… But it might happen!

• Direct substitution• substitute x for c• These functions are continuous at c

Properties of LimitsTheorem 1.1 Some Basic Limits

Let b and c be real numbers and let n be a positive integer.

1. 2. 3.

Example: Evaluating Basic Limits

a) b) c)

Theorem 1.2: Properties of Limits

Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

and

1. Scalar Multiple:

2. Sum or difference:

3. Product:

4. Quotient:

5. Power:

The Limit of a Polynomial

Theorem 1.3: Limits of Polynomial and Rational Functions

If p is a polynomial function and c is a real number, then

If r is a rational function given by r(x)=p(x)/q(x) and c is a real number such that q(c)≠0, then

The Limit of a Rational Function

Find the limit:

Theorem 1.4: The Limit of a Rational Function

Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c>0 if n is even.

Theorem 1.5: The Limit of a Composite Function

If f and g are functions such thatand , then

Limit of a Composite Functionf(x)=x2 + 4 and g(x)=

• Find

• Find

• Find

Theorem 1.6: Limits of Trigonometric Functions

Let c be a real number in the domain of the given trigonometric function.

1. 2.

3. 4.

5. 6.

Examples:

a.

b.

c.

Theorem 1.7: Functions that Agree at All But One Point

Let c be a real number and let f(x)=g(x) for all x≠c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and

Find the limit:

Dividing Out Technique Find the limit:

Rationalizing Technique

Find the limit:

Theorem 1.8: The Squeeze Theorem

If h(x) < f(x) < g(x) for all x in an open interval containing c, except possibly at c itself, and if

then exists and is equal to L.

Theorem 1.9: Two Special Trig Limits

1. 2.

Extra Example

A Limit Involving a Trig Function Find the limit:

A Limit Involving a Trig Function Find the limit:

Read 1.3Page 67 #17-75 every other odd, 85-89, 107, 108, 117-122

Classwork/Homework