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
