1.2 Finding Limits Graphically and Numerically, part 1 OBJECTIVE Estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist.

Consider this function. What happens to the value of f (x) when the value of x gets closer and closer and closer (but not necessarily equal) to 2? Read as the limit of f(x) as x approaches 2. We write this as: The answer can be found graphically, numerically, and analytically.

Consider this function. What happens to the value of f (x) when the value of x gets closer and closer and closer (but not necessarily equal) to 2? Read as the limit of f(x) as x approaches 2.

“the limit as x approaches a of f(x) is L.” DEFINITION As x approaches a, the f(x) approaches a certain value, L, called the limit. “the limit as x approaches a of f(x) is L.”

Finding the limit numerically

1) Consider the function x 4.9 4.99 4.999 4.9999 5 5.0001 5.001 5.01 5.1 g(x) 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.1 1 Basic idea: plug and chug and typically you’ll end up with a finite number. However, most limit problems don’t work out with plug and chug. Often you’ll get 0/0 or nonzero/0 So you need different techniques for finding the limit. Typically you’ll end up with a finite number

2) Consider the function x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f(x) 3.9 3.99 3.999 3.9999 und. 4.0001 4.001 4.01 4.1

3) Consider the function x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f(x) 11.41 11.94 11.994 11.999 und. 12.001 12.006 12.06 12.61

4) Consider the function x -4 -3 -2 -1 1 2 3 4 h(x) -1 -1 -1 -1 und. 1 1 1 1 One sided limits are in section 1.4 does not exist The roads don’t meet, so the limit d.n.e.

“the limit as x approaches 0 from the left of h(x) is -1.” One-side limits For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only. “the limit as x approaches 0 from the left of h(x) is -1.”

= 2 2 2 1 5) left hand limit right hand limit value of the function Also the limit as x approaches 1 has nothing to do with the value of f at 1. You can fill the circle, or move the closed circle and the limit is still going to be 2. At any point where the graph of f is continuous the y-coordinate will equal the limit of f as x approaches the x-coordinate at that point. So the value and limit coincide where ever the graph of f is continuous value of the function 1

2 1 2 6) does not exist left hand limit right hand limit The roads don’t meet, so the limit d.n.e. 6) does not exist left hand limit 2 right hand limit 1 There is no limit of f as x approaches 0. The lim as x →0 does not exist. However, we can describe the behavior of f near x = 0 in terms of one-sided limits. value of the function 2

1 1 2 7) = 1 left hand limit right hand limit value of the function There is no limit of f as x approaches 0. The lim as x →0 does not exist. However, we can describe the behavior of f near x = 0 in terms of one-sided limits. value of the function 2

1 1 8) d.n.e. left hand limit right hand limit value of the function The roads don’t meet, so the limit d.n.e. 8) d.n.e. left hand limit right hand limit 1 There is no limit of f as x approaches 0. The lim as x →0 does not exist. However, we can describe the behavior of f near x = 0 in terms of one-sided limits. value of the function 1

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Student Questionnaire (graded) Homework Read the syllabus Student Questionnaire (graded) Worksheet. Limits Worksheet A