1. 2.1 Introduction to Limits
Rita Korsunsky

2. Limit of a function •
Notation: • L
Intuitive Meaning: x a X

3. Example 1:
f(x) = x+2 x
f(x) = x + 2 3 2 1 = 3

4. Example 2 3 2 1 Even though g(1)  3, the limit is still 3 y = x +2
From example1= 3
Even though g(1)  3, the limit is still 3

5. One-sided Limits   Notation: y = f(x)
Intuitive Meaning: We can make f(x) as close to L as desired by choosing x sufficiently close to a, and x < a. 
f(x) L x a
Notation:
y = f(x) 
Intuitive Meaning: We can make f(x) as close to L as desired by choosing x sufficiently close to a, and x >a. 
f(x) L a x

6. Theorem
if and only if

7. Example 3
As the graph shows, f(x) does not approach a specific number L as x approaches 0 from the right and from the left, limit does not exist.

8. Example 4 y
D.N.E x

9. Example 5 y -1 1 x
D.N.E
(left-hand and right-hand limits are not equal)

10. Example 6 1 0 1 DNE 1 1 1 2 2 2 1 Given y = f(x), find the limits
from x = 0 to 4. 1
At x = 0:
At x = 1:
At x = 2:
At x = 3:
At x = 4:
0 1 DNE
Even though f(1) = 1
1 1 1
Even though f(2) = 2
2 2 2 1

11. Example 7

12. Example 8