MAT 125 – Applied Calculus 2.5 – One-Sided Limits & Continuity.

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Presentation transcript:

MAT 125 – Applied Calculus 2.5 – One-Sided Limits & Continuity

Today’s Class  We will be learning the following concepts today:  One-Sided Limits  Properties of Continuous Functions  The Intermediate Value Theorem Dr. Erickson 2.5 – One-Sided Limits & Continuity 2

One-Sided Limits Dr. Erickson 2.5 – One-Sided Limits & Continuity 3

One-Sided Limits Dr. Erickson 2.5 – One-Sided Limits & Continuity 4

When does a limit exist? – One-Sided Limits & Continuity Dr. Erickson

L c A general limit exists on f (x) when x = c if the right hand and left hand limits are both equal – One-Sided Limits & Continuity Dr. Erickson

Theorem 3 Dr. Erickson 2.5 – One-Sided Limits & Continuity 7

Dr. Erickson 2.5 – One-Sided Limits & Continuity 8

Example 1 – True or False Dr. Erickson 2.5 – One-Sided Limits & Continuity 9

Example 2  Find the indicated one-sided limit if it exists. Dr. Erickson 2.5 – One-Sided Limits & Continuity 10

Continuous Functions  Loosely speaking, a function is continuous at a point if the graph of the function at that point is devoid of holes, gaps, jumps, or breaks. Dr. Erickson 2.5 – One-Sided Limits & Continuity 11

Continuity of a Function at a Number Dr. Erickson 2.5 – One-Sided Limits & Continuity 12

Properties of Continuous Functions Dr. Erickson 2.5 – One-Sided Limits & Continuity 13

Properties of Continuous Functions Dr. Erickson 2.5 – One-Sided Limits & Continuity 14

Example 3  Find the values for x for which each function is continuous. Dr. Erickson 2.5 – One-Sided Limits & Continuity 15

Example 4  Find the values for x at which the function is discontinuous. Dr. Erickson 2.5 – One-Sided Limits & Continuity 16

Example 5 Dr. Erickson 2.5 – One-Sided Limits & Continuity 17

Example 6 Dr. Erickson 2.5 – One-Sided Limits & Continuity 18

Example 7 Dr. Erickson 2.5 – One-Sided Limits & Continuity 19

Theorem 4 – The Intermediate Value Theorem Dr. Erickson 2.5 – One-Sided Limits & Continuity 20

Example 8  Use the Intermediate Value Theorem to show that there exists a number c in the given interval such that f (c) = M. Dr. Erickson 2.5 – One-Sided Limits & Continuity 21

Example 9 Dr. Erickson 2.5 – One-Sided Limits & Continuity 22

Next Class  We will discuss the following concepts:  Slope of a Tangent Line  The Derivative  Differentiability & Continuity  Please read through Section 2.6 – The Derivative in your text book before next class. Dr. Erickson 2.5 – One-Sided Limits & Continuity 23