15.1 Notes

Analytical approach to evaluating limits of rational functions as x

approaches a finite number

15.1 Notes

Procedure for evaluating limits of rational functions where x approaches a number.

1. Attempt to evaluate the function at the value that x is approaching.

15.1 Notes – Example 1

Procedure for evaluating limits of rational functions where x approaches a number.

1. Attempt to evaluate the function at the value that x is approaching.

3

2lim 4xx x

6 3

2 2 4

15.1 Notes – Example 2

Procedure for evaluating limits of rational functions where x approaches a number.

1. Attempt to evaluate the function at the value that x is approaching.

This is called the indeterminate form.

2

4

16lim

4x

x

x

0

0

24 16

4 4

15.1 Notes

Procedure for evaluating limits of rational functions where x approaches a number.

1. Attempt to evaluate the function at the value that x is approaching.

2. If step #1 yields division by zero or the indeterminate form, attempt to factor the numerator and denominator and cancel common factors. Then evaluate the resulting function at the value x is approaching.

15.1 Notes – Example 22

4

16lim

4x

x

x

4lim 4xx

4

4 4lim

4x

x x

x

4 4 8

2

4

16lim 8

4x

x

x

15.1 Notes – Example 3

26

6lim

2 11 6x

x

x x

2

6 6

2 6 11 6 6

0

0

15.1 Notes – Example 3

26

6lim

2 11 6x

x

x x

6

1lim

2 1x x

6

6lim

2 1 6x

x

x x

1

2 6 1

1

13

26

6 1lim

2 11 6 13x

x

x x

15.1 Notes – Example 4

23

5lim

3x

x

x x 15

0

2

5 3

3 3 3

undefined

15.1 Notes – Example 4

23

5lim

3x

x

x x

5

3 3

3

5lim

3x

x

x x

3

5lim

3x x

5

0 undefined

15.1 Notes

Procedure for evaluating limits of rational functions where x approaches a number.

1. Attempt to evaluate the function at the value that x is approaching.

2. If step #1 yields division by zero or the indeterminate form, attempt to factor the numerator and denominator and cancel common factors. Then evaluate the resulting function at the value x is approaching.

3. If step #2 still yields division by zero or the indeterminate form, or the function doesn’t factor, or it factors but there aren’t any common factors, then the limit probably doesn’t exist. Show that the left-hand and right-hand limits approach infinity or negative infinity.

15.1 Notes – Example 4

23

5lim

3x

x

x x

DNE

3

5lim

3x x 3

5lim

3x x

15.1 Notes – Example 5

2

1

2 1lim

1x

x x

x

4

0

21 2 1 1

1 1

undefined

15.1 Notes – Example 52

1

2 1lim

1x

x x

x

1

1 1lim

1x

x x

x

4

0 undefined

1 1 1 1

1 1

15.1 Notes – Example 52

1

2 1lim

1x

x x

x

DNE

1

1 1lim

1x

x x

x

1

1 1lim

1x

x x

x

15.1 Notes

Procedure for evaluating limits of rational functions where x approaches a number.

1. Attempt to evaluate the function at the value that x is approaching.

2. If step #1 yields division by zero or the indeterminate form, attempt to factor the numerator and denominator and cancel common factors. Then evaluate the resulting function at the value x is approaching.

3. If step #2 still yields division by zero or the indeterminate form, or the function doesn’t factor, or it factors but there aren’t any common factors, then the limit probably doesn’t exist. Show that the left-hand and right-hand limits approach infinity or negative infinity.

15.1 Notes – Practice ProblemsEvaluate the limit.

1.

2.

3.

2

5

3 10lim

5x

x x

x

22

3lim

4x

x

x

2

2

7 10lim

2x

x x

x

15.1 Notes – Practice 1

2

2

7 10lim

2x

x x

x

22 7 2 10

2 2

2

2 5lim

2x

x x

x

28

0

28

0 2 2 2 5

2 2

15.1 Notes – Practice 12

2

7 10lim

2x

x x

x

DNE

2

2 5lim

2x

x x

x

2

2 5lim

2x

x x

x

15.1 Notes – Practice 2

2

5

3 10lim

5x

x x

x

25 3 5 10

5 5

5

2 5lim

5x

x x

x

0

0

2

5

3 10lim 7

5x

x x

x

5lim 2x

x

5 2

15.1 Notes – Practice 3

22

3lim

4x

x

x

2

2 3

2 4

2

3lim

2 2x

x

x x

1

0

2 3 1

2 2 2 2 0

15.1 Notes – Practice 3

22

3lim

4x

x

x

DNE

2

3lim

2 2x

x

x x

2

3lim

2 2x

x

x x