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1.5 Infinite Limits and 3.5 Limits at Infinity

1.5 Infinite Limits and 3.5 Limits at Infinity. AP Calculus I Ms. Hernandez (print in grayscale or black/white). AP Prep Questions / Warm Up. No Calculator! (a) 1 (b) 0 (c) e (d) – e (e) Nonexistent (a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE. AP Prep Questions / Warm Up.

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1.5 Infinite Limits and 3.5 Limits at Infinity

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  1. 1.5 Infinite Limitsand 3.5 Limits at Infinity AP Calculus I Ms. Hernandez (print in grayscale or black/white)

  2. AP Prep Questions / Warm Up No Calculator! (a) 1 (b) 0 (c) e (d) –e (e) Nonexistent (a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE

  3. AP Prep Questions / Warm Up No Calculator! (a) 1 (b) 0 (c) e (d) –e (e) Nonexistent (a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE

  4. 1.5 Infinite Limits • Vertical asymptotes at x=c will give you infinite limits • Take the limit at x=c and the behavior of the graph at x=c is a vertical asymptote then the limit is infinity • Really the limit does not exist, and that it fails to exist is b/c of the unbounded behavior (and we call it infinity)

  5. Determining Infinite Limits from a Graph • Example 1 pg 81 • Can you get different infinite limits from the left or right of a graph? • How do you find the vertical asymptote?

  6. Finding Vertical Asymptotes • Ex 2 pg 82 • Denominator = 0 at x = c AND the numerator is NOT zero • Thus, we have vertical asymptote at x = c • What happens when both num and den are BOTH Zero?!?!

  7. A Rational Function with Common Factors • When both num and den are both zero then we get an indeterminate form and we have to do something else … • Ex 3 pg 83 • Direct sub yields 0/0 or indeterminate form • We simplify to find vertical asymptotes but how do we solve the limit? When we simplify we still have indeterminate form.

  8. A Rational Function with Common Factors • Ex 3 pg 83: Direct sub yields 0/0 or indeterminate form. When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2. • Take lim as x-2 from left and right

  9. A Rational Function with Common Factors • Ex 3 pg 83: Direct sub yields 0/0 or indeterminate form. When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2. • Take lim as x-2 from left and right • Take values close to –2 from the right and values close to –2 from the left … Table and you will see values go to positive or negative infinity

  10. Determining Infinite Limits • Ex 4 pg 83 • Denominator = 0 when x = 1 AND the numerator is NOT zero • Thus, we have vertical asymptote at x=1 • But is the limit +infinity or –infinity? • Let x = small values close to c • Use your calculator to make sure – but they are not always your best friend!

  11. Properties of Infinite Limits • Page 84 • Sum/difference • Product L>0, L<0 • Quotient (#/infinity = 0) • Same properties for • Ex 5 pg 84

  12. Asymptotes & Limits at Infinity For the function , find (a) (b) (c) (d) (e) All horizontal asymptotes (f) All vertical asymptotes

  13. Asymptotes & Limits at Infinity For x>0, |x|=x (or my x-values are positive) 1/big = little and 1/little = big sign of denominator leads answer For x<0 |x|=-x (or my x-values are negative) 2 and –2 are HORIZONTAL Asymptotes

  14. Asymptotes & Limits at Infinity

  15. 3.5 Limit at Infinity • Horizontal asymptotes! • Lim as xinfinity of f(x) = horizontal asymptote • #/infinity = 0 • Infinity/infinity • Divide the numerator & denominator by a denominator degree of x

  16. Some examples • Ex 2-3 on pages #194-195 • What’s the graph look like on Ex 3.c • Called oblique asymptotes (not in cal 1) • KNOW Guidelines on page 195

  17. 2 horizontal asymptotes • Ex 4 pg 196 • Is the method for solving lim of f(x) with 2 horizontal asymptotes any different than if the f(x) only had 1 horizontal asymptotes?

  18. Trig f(x) • Ex 5 pg 197 • What is the difference in the behaviors of the two trig f(x) in this example? • Oscillating toward no value vs oscillating toward a value

  19. Word Problems !!!!! • Taking information from a word problem and apply properties of limits at infinity to solve • Ex 6 pg 197

  20. A word on infinite limits at infinity • Take a lim of f(x)  infinity and sometimes the answer is infinity • Ex 7 on page 198 • Uses property of f(x) • Ex 8 on page 198 • Uses LONG division of polynomials-Yuck!

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