Calculus and Analytical Geometry

1. Lecture # 6 Calculus and Analytical Geometry MTH 104

2. Limits at Infinity If the values of a variable x increases without bound, then we write and if the values of x decreases without bound, then we write For example As the denominator gets larger, the value of the fraction gets smaller.

3. Limits at Infinity An informal view If the values of f(x) eventually get as close as we like to a number L as x increases without bound, then we write Similarly, If the values of f(x) eventually get as close as we like to a number L as x increases without bound, then we write Lim as xinfinity of f(x) = horizontal asymptote

4. Limits at Infinity Example : This implies that there is horizontal asymptote at y=1

5. Infinite limits at Infinity An informal view

6. Infinite limits at Infinity Examples The end behavior of a polynomial matches the end behavior of its highest degree term.

7. Limits at Infinity Examples

8. Limits at Infinity

9. Limits at Infinity

10. 2 1 1 2 3 4 Continuity A function f is continuous at the point x = c if the following are true: This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

11. Continuity Example Determine wheather the following functions are continuous at x=2.

12. Continuity on an interval • A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: • F is continuous on (a, b). • f is continuous from the right at a. • f is continuous from the left at b. Example what can you say about the continuity of the function ? Natural domain of f(x) [-3, 3]

13. Continuity on an interval

14. Properties of continuous functions • Composites of continuous functions are continuous. • A polynomial is continuous every where. • A rational function is continuous at every point where the denominator is non zero, and has discontinuities at the points where the denominator is zero.

15. Intermediate Value theorem If f is continuous on a closed interval [a, b] and k is any number between f(a) and f(b), inclusive, then there is atleast at least one number x in the interval [a, b] such that f(x)=k.