13141973-11-Limits

13141973-11-Limits - Math 135 Business Calculus Spring 2009...

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Unformatted text preview: Math 135 Business Calculus Spring 2009 Class Notes 1.1 Limits: A Numerical and Graphical Approach The limit of a function is the fundamental concept in calculus and is used to define the derivative of a function, the subject of this first chapter. In this first section, we’ll introduce an intuitive definition of the limit of a function. Calculus is concerned with how function values, or outputs, change as the input changes. Suppose we have a function y = f ( x ). Then x is the input and y the output. Suppose that, as the input x gets closer and closer to some fixed number a , the function values get closer and closer to some fixed number L . The number L is called the limit of f as x approaches a . EXAMPLE Consider the function defined by f ( x ) = x 2 + x − 2 x − 1 . First note that f ( x ) is not defined at x = 1 since the denominator x − 1 equals 0 at x = 1. Even though the function is not defined at x = 1, we can still examine how the function behaves for x close to 1. We can do this either numerically by constructing a table of function values for values of x close to 1, or graphically by looking at its graph near x = 1. a) Complete the following table of values x approaches 1 from left −→ | ←− x approaches 1 from right x . 8 . 9 . 99 . 999 1 1.001 1.01 1.1 1.2 2 f ( x ) Based upon the table, what are the function values doing as...
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13141973-11-Limits - Math 135 Business Calculus Spring 2009...

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