Chapter 2 Limits and the Derivative

1. Chapter 2Limitsand theDerivative Section 2 Infinite Limits And Limits at Infinity

2. Infinite Limits Infinite limits and vertical asymptotes help describe the behavior of functions that are unbounded near x = a. We illustrate the concept of infinite limits using the function,

3. Example: Infinite Limit The graph of f(x) is shown. The graph indicates that there is no real number L that the values of f(x) approach as x approaches 1. The dashed vertical line at x = 1 illustrates the infinite limit as x approaches 1 from the right and left. Such a line is called a vertical asymptote.

4. Example: Infinite Limit In Table 1, as x approaches 1 from the right (values greater than 1), the function values grow positively without bound. In Table 2, as x approaches 1 from the left (values less than 1), The function values decrease negatively without bound.

5. Example: Infinite Limit The function is discontinuous at x = 1. (f(1) does not exist.) As x approaches 1 from the right, the values of f (x) are positive and become increasingly large. We say that f (x) increases without bound. Since ∞ is not a real number, this limit does not exist.

6. Example: Infinite Limit As x approaches 1 from the left, the values of f (x) are negative and become increasingly large in a negative sense. We say that f (x) decreases through negative values without bound. Since –∞ is not a real number, this limit does not exist.

7. The Infinity Symbol The infinity symbol ∞ is used to describe the manner in which these limits fail to exist. Such limits are called infinite limits. The symbol ∞ is used to describe positive growth without bound and the symbol –∞ is used to describe negative growth without bound.

8. Definition Infinite Limits and Vertical Asymptotes

9. Locating Vertical Asymptotes Polynomial functions have no vertical asymptotes. A vertical asymptote of a rational function can occur only at a zero of its denominator.

10. Theorem 1 Locating Vertical Asymptotes of Rational Functions

11. Example: Locating Vertical Asymptotes Describe the behavior of f at each zero of the denominator. Use ∞ or –∞ when appropriate. Solution: The denominator factors, x2 – 1 = (x – 1)(x + 1) and has two zeroes, x = –1 and x = 1. When x = –1, the numerator evaluates to –2 which is non-zero. According to Theorem 1, the line x = –1 is a vertical asymptote.

12. Example: Locating Vertical Asymptotes (continued) Solution: For value, x = 1, (which is a denominator zero), the numerator evaluates to 0 This gives the indeterminate 0/0 form and Theorem 1 does not apply. We use algebraic simplification to investigate the behavior of the function at x = 1.

13. Example: Locating Vertical Asymptotes (continued)

14. Limits at Infinity Limits at infinity and horizontal asymptotes are used to describe the behavior of functions as x assumes arbitrarily large positive values or arbitrarily large negative values. The symbol ∞ is used to indicate that an independent variable is increasing without bound through positive values. The symbol –∞ is used to indicate that an independent variable is decreasing without bound through negative values.

15. Theorem 2 Limits of Power Functions at Infinity

16. Example: Limit of a Polynomial Function at Infinity Let p(x) = 5x3– 2x2 – 3x + 6. Find the limit of p(x) as x approaches ∞ and as x approaches –∞.

17. Example: Limit of a Polynomial Function at Infinity (continued) The behavior of p(x) for large values is the same as the behavior of the highest degree term, 5x3.

18. Theorem 3 Limits of Polynomial Functions at Infinity A polynomial of degree 0 is a constant function p(x) = a0, and its limit as x approaches ∞ or –∞ is the number a0. Polynomial functions of degree 1 or greater never have horizontal asymptotes.

19. End Behavior of a Function The first of the two limits describes the right end behavior and the second describes the left end behavior.

20. Example: Find the End Behavior of a Function Give a pair of limit expressions that describe the end behavior of the function p(x) = 7x3 – 2x2 + 3x – 4.

21. Theorem 4 Limits of Rational Functions at Infinity/Horizontal Asymptotes of Rational Functions

22. Example: Finding Horizontal Asymptotes Find all horizontal asymptotes, if any, of the function. The function f(x) has no horizontal asymptotes.

23. Example: Finding Horizontal Asymptotes Find all horizontal asymptotes, if any, of the function. The x axis (y = 0) is a horizontal asymptote of f(x).

24. Example: Finding Horizontal Asymptotes Find all horizontal asymptotes, if any, of the function.

25. Example: Find the End Behavior of a Function A newly released smart-phone operating system gives users an update notice when they download a new app.

26. Example: Find the End Behavior of a Function A newly released smart-phone operating system gives users an update notice when they download a new app.