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"Calculus and its applications: 1.1" - Limits A Numerical and Graphical Approach have objective: Find limits of functions, if they exist, using numerical or graphical methods. | Limits: A Numerical and Graphical Approach OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods. DEFINITION: As x approaches a, the limit of f (x) is L, written if all values of f (x) are close to L for values of x that are sufficiently close, but not equal to, a. 1.1 Limits: A Numerical and Graphical Approach Slide 1.1- 2012 Pearson Education, Inc. All rights reserved THEOREM: As x approaches a, the limit of f (x) is L, if the limit from the left exists and the limit from the right exists and both limits are L. That is, if 1) and 2) then 1.1 Limits: A Numerical and Graphical Approach Slide 1.1- 2012 Pearson Education, Inc. All rights reserved 1.1 Limits: A Numerical and Graphical Approach Quick Check 1 Let What is ? What is the limit of as approaches ? Slide 1.1- 2012 Pearson Education, Inc. All rights reserved 1.1 Limits: A Numerical and Graphical Approach Quick Check 1 Solution a) 1.) Since , we will substitute in for , giving us the new equation 2.) Solving for , we get Thus does not exist. Slide 1.1- 2012 Pearson Education, Inc. All rights reserved 1.1 Limits: A Numerical and Graphical Approach Quick Check 1 Solution b) First let approach from the left: Thus it appears that is . Next let approach from the right: Thus it appears that is . Since both the left-hand and right-hand limits agree, . Slide 1.1- 2012 Pearson Education, Inc. All rights reserved Example 1: Consider the function H given by Graph the function, and find each of the following limits, if they exist. When necessary, state that the limit does not exist. a) 1.1 Limits: A Numerical and Graphical Approach b) Slide 1.1- 2012 Pearson Education, Inc. All rights reserved a) Limit Numerically First, let x approach 1 from the left: Thus, it appears that 0 0.5 0.8 0.9 0.99 0.999 H(x) 1.1 Limits: A Numerical and Graphical Approach 2 3 3.6 3.8 3.98 3.998 Slide 1.1- 2012 Pearson Education, Inc. All rights reserved a) Limit Numerically . | Limits: A Numerical and Graphical Approach OBJECTIVE Find limits of functions, if they exist, using numerical or graphical methods. DEFINITION: As x approaches a, the limit of f (x) is L, written if all values of f (x) are close to L for values of x that are sufficiently close, but not equal to, a. 1.1 Limits: A Numerical and Graphical Approach Slide 1.1- 2012 Pearson Education, Inc. All rights reserved THEOREM: As x approaches a, the limit of f (x) is L, if the limit from the left exists and the limit from the right exists and both limits are L. That is, if 1) and 2) then 1.1 Limits: A Numerical and Graphical Approach Slide 1.1- 2012 Pearson Education, Inc. All rights reserved 1.1 Limits: A Numerical and Graphical Approach Quick Check 1 Let What is ? What is the limit of as approaches ? Slide 1.1- 2012 Pearson Education, Inc. All rights reserved 1.1 Limits: A Numerical and Graphical Approach Quick Check 1 Solution a) 1.) Since , we will substitute in for , giving us the .
