1.5 Limits

Consider the value of a function f(x) as x approaches c, where c may or may not be in the domain of f(x)

Definition of limit

If f(x) gets closer and closer to a number L as x gets closer and closer to c from both sides, then L is the limit of f(x) as x approaches c. The behavior is expressed by writing:

Consider  f(x) = 3x + 1    as x gets very close to 4 which IS in the domain of f(x)

Let’s use a table with values approaching 4 from left and right

3.5

3.8

3.9

3.99

3.999

…

4

…

4.001

4.01

4.1

4.2

4.5

Consider 

What is  where 9 is NOT in the domain of the function

Use table:

8.5

8.8

8.9

8.99

8.999

…

9

…

9.001

9.01

9.1

9.2

9.5

From the table it appears that  ₌

Let's also examine the graph. “Hole in graph”

Find the limit using algebra:

The LIMIT is the behavior NEAR a point.

Algebraic Properties of Limits

If

Limits of Two Linear Functions For any constant k,

If these limits are used in conjunctions with the properties above, polynomial functions have limits f(c) when x approaches c. Rational functions have limits f(c) when x approaches c if f(c) is not equal to zero.

Examples:

If _, find

a) b)

c)

Find:

Limits Involving Infinity

“long term” or “end” behavior of a function

Limits at Infinity If the values of the function f(x) approach the number L as x increases

without bound, then

And similarly, when the functional values of f(x) approaches the number m

as x decreases without bound.

Examples

Find If the limit is infinity, indicate positive or negative.

#28 #32

#34 #36

Reciprocal Power Rules If A and k are constants with is defined for all x, then

Procedure for Evaluating a Limit at Infinity of

1. Divide each term of the quotient by the highest power of that appears in the denominator q(x)

2. Compute the resulting limit using algebraic properties of limits and the reciprocal rules.

Revisit #32-36 above.

#32

#34 #36

Assignment: 1.5; pgs 73-75; 1-35 odd, 43-49 odd.