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LIMITS AND LIMITS AT INFINITY Limit Review 1
23

Limits and Limits at infinity

Feb 22, 2016

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Limits and Limits at infinity. Limit Review 1. Limits can be calculated 3 ways. Numerically Graphically Analytically (direct substitution) Properties of limits (pages 56-58). Graphically. Determine the value of y as x approaches some number c . Formal Definition of a Limit. - PowerPoint PPT Presentation
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Page 1: Limits and Limits at infinity

LIMITS AND LIMITS AT INFINITY

Limit Review 1

Page 2: Limits and Limits at infinity

Limits can be calculated 3 ways

• Numerically

• Graphically

• Analytically (direct substitution)

• Properties of limits (pages 56-58)

right theandleft both the fromnumber sameapproach must

Page 3: Limits and Limits at infinity

Graphically

• Determine the value of y as x approaches some number c

Page 4: Limits and Limits at infinity

Formal Definition of a Limit

Page 5: Limits and Limits at infinity

Analytic techniques

• Direct substitution • Factoring- cancel common factors• Rationalize- multiply by the conjugate

Page 6: Limits and Limits at infinity

1) Find

2) Find

434lim 2

2

4

xxxx

x

432lim 2

5

xx

x

Page 7: Limits and Limits at infinity

3) Find

732lim

7

xx

x

Page 8: Limits and Limits at infinity

One sided limits

• left

• Right

• Limit existence

cxxfcx

)(lim

cxxfcx

)(lim

)(lim )(lim xfxfcxcx

Page 9: Limits and Limits at infinity

4) Find xx

x 0lim

Page 10: Limits and Limits at infinity

5) Find 2

2lim2

2

xxx

x

2222

2xxxx

x

Page 11: Limits and Limits at infinity

Infinite Limits• Limit of f(x) increases or decreases without

bound as x approaches some number c

• Limit does not exist DNE• Vertical asymptotes occur when limits are

infinite• Limit as x approaches 1

or

Page 12: Limits and Limits at infinity

6) Find

limx 1

?x 1

Page 13: Limits and Limits at infinity

7) Find 1

2 lim1

x

xx

Page 14: Limits and Limits at infinity

8) Find xx

1 xlim 2

0

Page 15: Limits and Limits at infinity

9) Find 344

16 lim 2

2

21

xxxx

x

Page 16: Limits and Limits at infinity

Limits at Infinity

• Identifies the behavior of the function as x tends to infinity

• If , then L is a horizontal asymptote

• Usually f(x) is rational• Limit as x approaches infinity

Lxfx

)(lim

Page 17: Limits and Limits at infinity

10) Find 2

25 limxx

Page 18: Limits and Limits at infinity

11) Find 32

6 lim xx

x

Page 19: Limits and Limits at infinity

Limits at Infinity Shortcuts

• If the degree is higher in the denominator, then the limit is equal to zero

• If the degree is higher in the numerator then the limit does not exist

• If the degrees are the same, then the limit equals the ratio of the coefficients

BOBO BOTN EATS DC

Page 20: Limits and Limits at infinity

12) Find

13) Find

35 lim

2

xx

x

12 lim

2 x

xx xrecall 2x

Page 21: Limits and Limits at infinity

14) Find and all asymptotes 4

82 lim 2

2

xxx

x

Page 22: Limits and Limits at infinity

15) Find and all asymptotes 11 lim 2

2

xx

x

Page 23: Limits and Limits at infinity

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