1. Idea of Limit
The limit of a function at a point is based on "local" behavior of the function near that point.
It is the value that the function would have if we consider the values at nearby points, but not at the point itself.
y=f(x) x

2. Example Last time we looked at formulas like
This is defined at every point except x = 1.
What should its value be there?
For this we take limits.

3. Look at a table values x f(x) 1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001
1.0001
2.0001 1
undefined
.9999
1.9999
.999
1.999
.99
1.99 .9
1.9

4. Look at graph
y=2
x=1

5. Notation
The reads:
The limit of f(x), as x approaches a, is equal to L
Meaning:
As x gets closer to a, f(x) gets closer to L.

6. Limits of simple functions
Constant functions have limits. The limit of the value is the constant value.
Straight line functions have limits equal to the value at the limit point.
Polynomials have limits equal to their value at the limit point.
For these functions:

7. Polynomial examples
Find

8. More kinds of functions

9. Examples using algebra
Find limit of f(x)=(x-2)/(x2-4) as x approaches 2.

10. The limit does not always exist
The limit does not always exit.
Example:

11. Fancier Example

12. More functions f(x)=sin(x)/x f(x)=sin(π/x)
Heaviside function – H(t) is zero for t less than zero and one for t greater than or equal to 0.

13. One sided limits: left Limit from left: This reads:
The left-hand limit of f(x), as x approaches a, is equal to L. or
The limit of f(x), as x approaches a from the left, is equal to L.

14. One sided limits: right
Limit from right:
This reads:
The right-hand limit of f(x), as x approaches a, is equal to L. or
The limit of f(x), as x approaches a from the right,
is equal to L.

15. Examples
Square roots

16. Limit of infinity This reads:
The limit of f(x), as x approaches a, is equal to .
It means that as x approaches a, f(x) get larger.

17. Limit of minus infinity
This reads:
The limit of f(x), as x approaches a, is equal to .
It means that as x approaches a, f(x) get larger.

18. Examples
f(x)=1/x2
More?

19. Vertical asymptotes Special limits
When these any of these happen the line x=a is called a vertical asymptote.

20. Examples
Various rational functions to match the cases.