Chap2_Sec2 - LIMITS LIMITS 2 2.2 The Limit of a Function...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: LIMITS LIMITS 2 2.2 The Limit of a Function LIMITS In this section, we will learn: About limits in general and about numerical and graphical methods for computing them. Let’s investigate the behavior of the function f defined by f ( x ) = x 2 – x + 2 for values of x near 2. The following table gives values of f ( x ) for values of x close to 2, but not equal to 2. THE LIMIT OF A FUNCTION From the table and the graph of f (a parabola) shown in the figure, we see that, when x is close to 2 (on either side of 2), f ( x ) is close to 4. THE LIMIT OF A FUNCTION In fact, it appears that we can make the values of f ( x ) as close as we like to 4 by taking x sufficiently close to 2. THE LIMIT OF A FUNCTION We express this by saying “the limit of the function f ( x ) = x 2 – x + 2 as x approaches 2 is equal to 4.” The notation for this is: ( 29 2 2 lim 2 4 x x x →- + = THE LIMIT OF A FUNCTION In general, we use the following notation. We write and say “the limit of f ( x ), as x approaches a, equals L ” if we can make the values of f ( x ) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a ) but not equal to a . ( 29 lim x a f x L → = THE LIMIT OF A FUNCTION Definition 1 Roughly speaking, this says that the values of f ( x ) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a ) but x a . A more precise definition will be given in Section 2.4. THE LIMIT OF A FUNCTION ≠ An alternative notation for is as which is usually read “ f ( x ) approaches L as x approaches a .” ( 29 lim x a f x L → = THE LIMIT OF A FUNCTION ( ) f x L → x a → Notice the phrase “but x a ” in the definition of limit. This means that, in finding the limit of f ( x ) as x approaches a , we never consider x = a . In fact, f ( x ) need not even be defined when x = a . The only thing that matters is how f is defined near a . THE LIMIT OF A FUNCTION ≠ The figure shows the graphs of three functions. Note that, in the third graph, f ( a ) is not defined and, in the second graph, . However, in each case, regardless of what happens at a , it is true that . THE LIMIT OF A FUNCTION ( ) f x L ≠ lim ( ) x a f x L → = 2 1 1 lim 1 x x x →-- THE LIMIT OF A FUNCTION Example 1 lim ( ) x a f x → Guess the value of . The tables give values of f ( x ) (correct to six decimal places) for values of x that approach 1 (but are not equal to 1). On the basis of the values, we make the guess that THE LIMIT OF A FUNCTION Example 1 2 1 1 lim 0.5 1 x x x →- =- Example 1 is illustrated by the graph of f in the figure....
View Full Document

This note was uploaded on 03/04/2011 for the course MAT 3401 taught by Professor Munther during the Spring '11 term at Cal Poly Pomona.

Page1 / 65

Chap2_Sec2 - LIMITS LIMITS 2 2.2 The Limit of a Function...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online