1. 2.4 – Definition for Limits
Math 1304 Calculus I
2.4 – Definition for Limits

2. Recall Notation for Limits
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.

3. Questions Can we make it more precise?
Can we use this more precise definition to prove the rules?

4. Distance
What do we mean by “As x gets closer to a, f(x) gets closer to L”.
Can we measure how close? (distance)
What’s the distance?
|x-a| is distance between x and a
|f(x)-L| is distance between f(x) and L x a
f(x) L

5. Rules for Distance
If a and b are real number |a-b| is the distance between them.
The distance between two different numbers is positive.
If the distance between two numbers is zero, then they are equal.
If a, b, c are real numbers, then
|a - c| ≤ |a - b| + |b - c|

6. How close? Measuring distance: argument and value.x a
f(x) L
Distance of arguments =|x - a|
Distance of values = |f(x)–L|
Can we make the distance between the values of f and L small by making the distance between x and a small?
Turn this into a bargain:
Given any >0, find >0 such that
|f(x)-L|< , whenever 0<|x-a|< .

7. Formal Definition of Limits
Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is L, if for each positive real number >0 there is a positive real >0 such that
|f(x)-L|< , whenever 0<|x-a|< .
When this happens we write

8. Other ways to say it Given >0 there is >0 such that
|f(x)-L|< , whenever 0<|x-a|< .
0<|x-a|<   |f(x)-L|< 

9. Definition in terms of intervals
Given >0 there is >0 such that
a- < x < a+ and x≠a  L- < f(x) < L+   
a-  a
a+  
L - L
L+

10. Picture
L+
f(x) L
L-
a- a x
a+

11. Example
Using the above definition, prove that f(x) = 2x+1 has limit 5 at x=2
Method: work backwards: compute and estimate the distance |f(x)-L| in terms of the distance |x-a|
Use the estimate to guess a delta, given the epsilon.

12. Nearby Behavior
Note that if two functions agree except at a point a, they have the same limit at a, if it exists.
Stronger result: If two functions agree on an open interval around a point a, but not necessarily at a, and one has a limit at a, then they have the same limit at a.

13. Proof of Rules Can prove the above rules from this definition.
Example: the sum rule (in class)
Note: we need the triangle inequality:
|A+B|  |A|+|B|

14. One-sided limits: Left
Definition of left-hand limit
if for each positive real number >0 there is a positive real >0 such that |f(x)-L| < , whenever a-<x<a.

15. One-sided limits: Right
Definition of left-hand limit
if for each positive real number >0 there is a positive real >0 such that |f(x)-L| < , whenever a<x<a+.

16. Limits of plus infinity
Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is , if for each real number M there is a positive real >0 such that f(x)>M, whenever 0<|x-a|< . When this happens we write

17. Limits of minus infinity
Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. We say that the limit of f(x) as x approaches a is -, if for each real number M there is a positive real >0 such that f(x)<M, whenever 0<|x-a|< . When this happens we write