"Calculus and its applications: " - 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. Limits: A Numerical and Graphical Approach Slide 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 Limits: A Numerical and Graphical Approach Slide 2012 Pearson Education, Inc. All rights reserved Limits: A Numerical and Graphical Approach Quick Check 1 Let What is ? What is the limit of as approaches ? Slide 2012 Pearson Education, Inc. All rights reserved 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 2012 Pearson Education, Inc. All rights reserved 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 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) Limits: A Numerical and Graphical Approach b) Slide 2012 Pearson Education, Inc. All rights reserved a) Limit Numerically First, let x approach 1 from the left: Thus, it appears that 0 H(x) Limits: A Numerical and Graphical Approach 2 3 Slide 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. Limits: A Numerical and Graphical Approach Slide 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 Limits: A Numerical and Graphical Approach Slide 2012 Pearson Education, Inc. All rights reserved Limits: A Numerical and Graphical Approach Quick Check 1 Let What is ? What is the limit of as approaches ? Slide 2012 Pearson Education, Inc. All rights reserved Limits: A Numerical and Graphical Approach Quick Check 1 Solution a) 1.) Since , we will substitute in for , giving us the .
