MAT 1234 Calculus I Section 2.3 Part I Using the Limit Laws

Quiz Tomorrow and … Quiz :1.5, 1.6I Homework 1.6 Part I Do your HW ASAP. Write out your solutions carefully in a notebook - You want to have a reference before the exams…and bonus points for your first exam Tutoring is available!!!

Recall Limit of the following form is important 1.4: Estimate limits by tables 1.6: Compute limits by algebra 1.5: Formally define limits

Preview Limit Laws Direct Substitution Property Practical summary of all the limit laws

Limit Laws 11 limit laws that “help” us to compute limits. Foundation of computing limits, but tedious to use. Practical methods will be introduced.

Limit Laws 7.

Limit Laws 8.

Limit Laws If and exist, then

Example 1

Limit Laws 1. If and exist, then Why the assumption above is important?

Example Find

Direct Substitution Property If f(x) is a polynomial, then Also true if f(x) is a rational function and a is in the domain of f

Direct Substitution Property If f(x) is a polynomial, then Also true if f(x) is a rational function and a is in the domain of f

Direct Substitution Property If f(x) is a polynomial, then Also true if f(x) is a rational function and a is in the domain of f

Why? Polynomials are “continuous” functions x y a

Why? Polynomials are “continuous” functions

Example 1 (Polynomial)

Remark 1 Once you substitute in the number, you do not need the limit sign anymore.

Example 2 (Rational Function, a in the domain) 3 is in the domain of the rational function

Example 2 (Rational Function, a in the domain) 3 is in the domain of the rational function

Direct Substitution Property Can be extended to other functions such as n-th root. Not for all functions such as absolute value, piecewise defined functions.

Limit Laws Summary Use Direct Substitutions if possible*. That is, plug in x=a when it is defined. * Sums, differences, products, quotients, n-th root functions of polynomials,

Example 3

Q&A Q: What to do if the answer is undefined when plugging in x=a ? A: Try the following techniques

Example 4 (Simplify)

1.Use equal signs 2.Use parentheses for expressions with sums and differences of more than 1 term. 3. Show the substitution step. Reminders

4. Do not actually “cross out” terms.

Remark 1 Again Once you substitute in the number, you do not need the limit sign anymore.

Example 5 (Combine the terms)

Remark 1 Again (What? Again!) Once you substitute in the number, you do not need the limit sign anymore.

Example 7 (Multiply by conjugate)

Review of conjugates The conjugate of is The product of conjugates is

Example 7 (Multiply by conjugate)

Review: We learned… Limit Laws Direct Substitution Property of polynomials and rational functions Techniques Simplify Combine the terms Multiply by conjugate

Classwork Use pencils Use “=“ signs Do not “cross out” anything. Do not skip steps Once you substitute in the number, you do not need the limit sign anymore.