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 rules to compute limits. You can think of them 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 is the limit from the left.
It means:
As x gets closer to a from left, f(x) gets closer to L.

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

6. Theorem on Left and Right Limits
Theorem: The limit exists if and only if the left and right limits both exist and are equal.

7. 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.

8. 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:

9. 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.

10. List of Rules in the Book
See page 99 and101 for a set of rules for limits along these lines.
Rules 1, 2, 4, and 5 are what we just discussed for the operations (+, -, x, /).
Rules 7 (limit of a constant function) and 8 (the identity function) are basic starting points.
Rules 6 and 11 cover powers and roots.
Rules 9 and 10 are special cases of powers and roots.

11. Examples
In class – like some hw problems

12. Limits and Inequalities
See and study Theorems 2 and 3 on page 105.
Theorem 2 says how limits commute with inequalities (note: inequalities are not strict).
Theorem 3 is the Squeeze Theorem.