MAT 1221 Survey of Calculus Section 1.5 Limits

Seating - Group Activities You are going to pair up with a partner to work on classwork problems. No communications between any groups. Any two groups should have at least one empty seat between them.

Quiz You have 10 min. for this quiz. If you are done early, please wait and do not make noises.

Reminder WebAssign Homework 1.5 Quiz 01 on Thursday Read the next section on the schedule (From now on, you always read for the next class, I will not remind you again)

What do we care? How fast “things” are going The velocity of a particle The “speed” of formation of chemicals The rate of change of a population

A Leaking Tank

Fact: Slope of Tangent Line

First Important Task Develop the theory of finding the slope of the tangent lines (Derivatives)

Tangent Lines Tangent lines may not exist

Limits We want to mathematically distinguish the 2 curves

1.5 Preview One-sided limits The existence of limits Continuous at a point Use algebra to compute limits

Left-Hand Limit x y 3 2 The left-hand limit is 2 when x approaches 3 Notation: Independent of f(3) y=f(x)

Left-Hand Limit x y 3 2 y=f(x)

Left-Hand Limit x y 3 2 y=f(x)

Left-Hand Limit x y 3 2 y=f(x)

Left-Hand Limit** (Formal) We write and say “the left-hand limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to a and x less than a.

Right-Hand Limit x y 2 4 The right-hand limit is 4 when x approaches 2 Notation: Independent of f(2) y=f(x)

Right-Hand Limit x y 2 4 y=f(x)

Right-Hand Limit x y 2 4 y=f(x)

Right-Hand Limit x y 2 4 y=f(x)

Right-Hand Limit** (Formal) We write and say “the right-hand limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to a and x greater than a.

Limit of a Function

Example 1 x y y=f(x)

Example 2 x y y=f(x) 2.5

Infinite Limits x y a y=f(x) The left-hand limit DNE Notation: is not a number

Infinite Limits Other possible situations are similar.

Continuous at a Point

Definition is defined exists

Example 3 (example 1 above) x y y=f(x)

Example 3 (example 1 above) x y y=f(x)

So,…

Continuous Functions It can be proved that Polynomials continuous everywhere

Example 4 (Polynomial)

Remark Use equal signs Show the substitution step Once you substitute the number, you do not need the limit sign anymore.

Example 5 (Rational Function)

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

Example 6

Limit Operations There are a lot of limit operations listed in the book. If and exist, then

Limit Operations Summary

Example 7 (Simplification)

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

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 8 (Multiply by conjugate)

Review of conjugates

Example 8 (Multiply by conjugate)

Hint: Do NOT expand the denominator!

Classwork!

Expectations You should have the “lim” notations until you can plug in the number. You should have equal signs. Do not skip steps. When ask of, you need to start your solutions with or

Classwork Works with one partner and one partner ONLY. Use pencils, please. Caro and I will give you hints if you need help. As usual, you are supposed to know your algebra. You can go after One of us check BOTH of your work.

Classwork Problem 1 Let us start (a) together