Limit of functions
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1) One-sided limits describe the value a function approaches as the input gets closer to a number from the left or right. 2) The limit of a function exists if and only if the one-sided limits are equal as the input approaches the number. 3) Limits at infinity describe the value a function approaches as the input increases without bound toward positive or negative infinity.
Limit of functions
- 1. LIMIT OF FUNCTIONS One-sided limit, limit, existence of limit, limit at infinity, infinite limit
- 2. One-sided limit Given a function defined by 𝑦 = 𝑓 𝑥 lim 𝑥→𝑎+ 𝑓 𝑥 Is the value y (real number) being approached by f(x) as x gets closer and closer to a from the right.
- 3. One-sided limit Given a function defined by 𝑦 = 𝑓 𝑥 lim 𝑥→𝑎− 𝑓 𝑥 Is the value y (real number) “being approached” by f(x) as x gets closer and closer to a from the left.
- 4. One-sided limit Consider the function defined by a. Evaluate lim 𝑥→−3− 𝑓 𝑥 b. Evaluate lim 𝑥→−3+ 𝑓 𝑥
- 5. One-sided limit 𝑓 𝑥 = 𝑥2 − 9 𝑥 + 3 = 𝑥 + 3 𝑥 − 3 𝑥 + 3 = 𝑥 − 3, 𝑥 ≠ −3 𝐷𝑓 = −∞, −3 ∪ −3, ∞ 0 3 0 3
- 6. One-sided limit 3 92 x x xf 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
- 7. One-sided limit 3 92 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 𝑥→𝑎+ 𝑓 𝑥 = lim 𝑥→𝑎− 𝑓 𝑥
- 8. One-sided limit Given a function defined by 𝑓 𝑥 = 𝑥2 − 9 𝑥 + 3 lim 𝑥→−3− 𝑓 𝑥 = lim 𝑥→−3+ 𝑓 𝑥 lim 𝑥→−3 𝑓 𝑥 = −6
- 9. One-sided limit Functions whose limit at a does not exist. Example 2.1 Evaluate lim 𝑥→1 𝑓 𝑥 , does it exist?
- 10. One-sided limit a. Evaluate lim 𝑥→1− 𝑓 𝑥 and lim 𝑥→1+ 𝑓 𝑥 . b. Are they equal? Conclusion: the limit does not exist
- 11. One-sided limit Function whose limit does not exist at a. Example 2.4 Evaluate lim 𝑥→3 𝑓 𝑥 where 𝑓 𝑥 = 3𝑥 + 9 𝑥2 − 9
- 12. One-sided limit The limits do not point to a specific real number. Conclusion: limit does not exist as x approaches 3
- 13. Limit Theorems
- 14. Limit at infinity Evaluate lim 𝑥→∞ 𝑓 𝑥 and lim 𝑥→−∞ 𝑓 𝑥 , where
- 15. Limit at infinity As x approaches positive infinity or negative infinity, the quotient approaches zero.
- 16. Limit at infinity Evaluate lim 𝑥→∞ 𝑓 𝑥 where Solution
- 17. Limit at Infinity and Horizontal Asymptote Obtain the horizontal asymptote of Solution
- 18. Limit at infinity and horizontal asymptote Horizontal Asymptote 𝑦 = −1
- 19. Limit at infinity and asymptotes (horizontal and oblique)
- 20. Limit at infinity and asymptotes (horizontal and oblique) Examples
- 21. Limit at infinity and asymptotes (horizontal and oblique) Perform long division on
- 22. Limit of some trigonometric functions Theorems lim 𝑥→0 sin 𝑥 𝑥 = 1 lim 𝑥→0 cos 𝑥−1 𝑥 = 0