1. Math 1304 Calculus I
2.3 – Rules for Limits

2. Plan
So far (in section 2.1 and 2.2) we have explored the need for limits, notation, how they behave, and some examples.
Today in this section (2.3), we explore how to compute limits. You can think of this as axioms for limits.
In the next section we explore an actual definition of limits.

3. Recall Notation This means:
As x gets closer to a, f(x) gets closer to L.

4. Notation for left and right limits
This means:
As x gets closer to a from left, f(x) gets closer to L.

5. Notation for left and right limits
This means:
As x gets closer to a from right, f(x) gets closer to L.

6. Calculus of Limits Certain rules let us calculate limits.
What rules would be want?
Suppose we know the limit of two functions at a point. What would be the limit of various combinations of these functions?
Next slide shows some of these rules.

7. Rules for Limits
Sum Rule: At a given point, the limit of a sum of functions is the sum of the limits, provided that they both exist.
Difference Rule: At a given point, the limit of a difference of functions is the difference of the limits, provided that they both exist.
Scalar multiplier rule: At a given point, the limit of a constant times a functions is the constant times the limit of the function, provided that it exists.
Product Rule: At a given point, the limit of a product of functions is the product of the limits, provided that they both exist.
Quotient Rule: At a given point, the limit of a quotient of functions is the quotient of the limits, provided that they both exist and that the limit of the denominator is not zero.
Formulas:

8. Continuity=Direct Substitution
Recall: A function is said to be continuous at a point if its value at that point equals its limit at that point. (limxa f(x) = f(a))
Basic rule: Limits “commmute” with continuous operations. (limxa f(u)=f(limxa u)
Continuity of basic operations: addition, subtraction, multiplication, scalar multiplication, powers, roots, are all continuous. This is equivalent to the rules for limits.
Continuity and limits of constants, polynomials, rational functions, roots, etc. mean that limits also “commute” with these operations.

9. Examples
In class
Page 101

10. More Rules
See page 101 for rules 6-11 (go over in class)