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Limit of functions

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Limit of function BSCOE

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Limit of functions

  1. 1. LIMIT OF FUNCTIONS One-sided limit, limit, existence of limit, limit at infinity, infinite limit
  2. 2. One-sided limit Given a function defined by 𝑦 = 𝑓 π‘₯ lim π‘₯β†’π‘Ž+ 𝑓 π‘₯ Is the value y (real number) being approached by f(x) as x gets closer and closer to a from the right.
  3. 3. One-sided limit Given a function defined by 𝑦 = 𝑓 π‘₯ lim π‘₯β†’π‘Žβˆ’ 𝑓 π‘₯ Is the value y (real number) β€œbeing approached” by f(x) as x gets closer and closer to a from the left.
  4. 4. One-sided limit Consider the function defined by a. Evaluate lim π‘₯β†’βˆ’3βˆ’ 𝑓 π‘₯ b. Evaluate lim π‘₯β†’βˆ’3+ 𝑓 π‘₯
  5. 5. One-sided limit 𝑓 π‘₯ = π‘₯2 βˆ’ 9 π‘₯ + 3 = π‘₯ + 3 π‘₯ βˆ’ 3 π‘₯ + 3 = π‘₯ βˆ’ 3, π‘₯ β‰  βˆ’3 𝐷𝑓 = βˆ’βˆž, βˆ’3 βˆͺ βˆ’3, ∞ 0 3ο€­ 0 3ο€­
  6. 6. One-sided limit   3 92  ο€­ ο€½ x x xf x   3 92  ο€­ ο€½ x x xf x   3 92  ο€­ ο€½ 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
  7. 7. One-sided limit   3 92  ο€­ ο€½ x x xf Theorem 2.1 Existence of Limit The limit of a function exists if and only if the one- sided limits of the function are equal lim π‘₯β†’π‘Ž+ 𝑓 π‘₯ = lim π‘₯β†’π‘Žβˆ’ 𝑓 π‘₯
  8. 8. One-sided limit Given a function defined by 𝑓 π‘₯ = π‘₯2 βˆ’ 9 π‘₯ + 3 lim π‘₯β†’βˆ’3βˆ’ 𝑓 π‘₯ = lim π‘₯β†’βˆ’3+ 𝑓 π‘₯ lim π‘₯β†’βˆ’3 𝑓 π‘₯ = βˆ’6
  9. 9. One-sided limit Functions whose limit at a does not exist. Example 2.1 Evaluate lim π‘₯β†’1 𝑓 π‘₯ , does it exist?
  10. 10. One-sided limit a. Evaluate lim π‘₯β†’1βˆ’ 𝑓 π‘₯ and lim π‘₯β†’1+ 𝑓 π‘₯ . b. Are they equal? Conclusion: the limit does not exist
  11. 11. One-sided limit Function whose limit does not exist at a. Example 2.4 Evaluate lim π‘₯β†’3 𝑓 π‘₯ where 𝑓 π‘₯ = 3π‘₯ + 9 π‘₯2 βˆ’ 9
  12. 12. One-sided limit The limits do not point to a specific real number. Conclusion: limit does not exist as x approaches 3
  13. 13. Limit Theorems
  14. 14. Limit at infinity Evaluate lim π‘₯β†’βˆž 𝑓 π‘₯ and lim π‘₯β†’βˆ’βˆž 𝑓 π‘₯ , where
  15. 15. Limit at infinity As x approaches positive infinity or negative infinity, the quotient approaches zero.
  16. 16. Limit at infinity Evaluate lim π‘₯β†’βˆž 𝑓 π‘₯ where Solution
  17. 17. Limit at Infinity and Horizontal Asymptote Obtain the horizontal asymptote of Solution
  18. 18. Limit at infinity and horizontal asymptote Horizontal Asymptote 𝑦 = βˆ’1
  19. 19. Limit at infinity and asymptotes (horizontal and oblique)
  20. 20. Limit at infinity and asymptotes (horizontal and oblique) Examples
  21. 21. Limit at infinity and asymptotes (horizontal and oblique) Perform long division on
  22. 22. Limit of some trigonometric functions Theorems lim π‘₯β†’0 sin π‘₯ π‘₯ = 1 lim π‘₯β†’0 cos π‘₯βˆ’1 π‘₯ = 0
  23. 23. Limit of trigonometric functions
  24. 24. Limit of trigonometric functions
  25. 25. Limit of trigonometric functions
  26. 26. Limit of trigonometric functions

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