Limit of functions (CBA College of Business Ad)
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Limit of functions One-sided limit, limit, existence of limit, limit at infinity, infinite limit
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Transcript of Limit of functions (CBA College of Business Ad)
- 1. One-sided limit, limit, existence of limit, limit at infinity, infinite limit
- 2. One-sided limit xf ax lim Limit of f(x) as x approaches a from the left.
- 3. One-sided limit xf ax lim Limit of f(x) as x approaches a from the right.
- 4. One-sided limit Consider the function defined by ?lim 3 xf x ?lim 3 xf x
- 5. One-sided limit Where does f(x) point to as x approaches -3 from the left? ?lim 3 xf x ?lim 3 xf x Where does f(x) point to as x approaches -3 from the right?
- 6. One-sided limit ?lim 3 xf x ?lim 3 xf x
- 7. One-sided limit 3where 3 3 33 3 92 x x x xx x x xf
- 8. One-sided limit x 3 92 x x xf x 3 92 x x xf -3.1 -6.1 -2.9 -5.9 -3.01 -6.01 -2.99 -5.99 -3.001 -6.001 -2.999 -5.999 -3.0001 -6.0001 -2.9999 -5.9999 Table 2.1 Some numerical computations close to -3 from the left and right
- 9. One-sided limit 6lim 3 xf x 6lim 3 xf x
- 10. Existence of Limit Given a function defined by The limit of f as x approaches -3 exists and it is equal to -6 because xf x 3 lim 6lim 3 xf x
- 11. Existence of Limit 6lim 3 xf x
- 12. One-sided limit Functions whose limit at a does not exist. Example 2.1 Evaluate lim 1 , does it exist?Does the limit exists as x approaches 1?
- 13. One-sided limit Conclusion: the limit does not exist Use the criteria to prove existence of limit at x = 1
- 14. Existence of Limit
- 15. Existence of Limit
- 16. Existence of limit
- 17. Existence of Limit The limits do not point to a specific real number. Conclusion: limit does not exist as x approaches 3 x = 3 is the vertical asymptote
- 18. Existence of Limit L is a specific real number
- 19. Limit Theorems
- 20. Limit at infinity Carrying capacity
- 21. Limit at infinity As x approaches positive infinity or negative infinity, the quotient approaches zero.
- 22. Limit at infinityEvaluate lim where Solution
- 23. Limit at Infinity and Horizontal Asymptote Obtain the horizontal asymptote of Solution
- 24. Limit at infinity and horizontal asymptote Horizontal Asymptote = 1
- 25. Limit at infinity and asymptotes (horizontal and oblique)
- 26. Limit at infinity and asymptotes (horizontal and oblique) Examples
- 27. Limit at infinity and asymptotes (horizontal and oblique) Perform long division on
- 28. Limit of some trigonometric functions Theorems lim 0 sin = 1 lim 0 cos 1 = 0
- 29. Limit of trigonometric functions
- 30. Limit of trigonometric functions
- 31. Limit of trigonometric functions
- 32. Limit of trigonometric functions