Overview - we get closer and closer to the y-value 4, we...

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What do we mean by limit ? Compare our 2 4 ) ( 2 = x x x f and 2 5 ) ( 2 = x x x g Notice both are undefined at x = 2 The functions are shown below. g(x)
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Let's look at the y-values on either side of 2 for both of these functions: h Picking numbers smaller than 2 ( x approaches 2 from the left-hand side) h Picking numbers larger than 2 ( approaches 2 from the right-hand side):
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We use the notation 2 x to mean x approaches 2 from values on the left side of 2 + 2 x to mean x approaches 2 from values on the right We say: 4 ) ( lim 2 = + x f x because as we get very very very VERY close to 2 from the right, we approach 4. 4 ) ( lim 2 = x f x because as we get very very very VERY close to 2 from the left, we approach 4. Since as we get very, very close to 2 from BOTH sides,
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Unformatted text preview: we get closer and closer to the y-value 4, we can say 4 ) ( lim 2 = x f x When there is no + or on the number x is approaching, it means when x approaches the number from BOTH sides. For g(x) ) ( lim 2 x g x + does not exist because the y-values decrease without bound. We do not have a limit for ) ( x g at 2, but we may describe the behavior by looking at each side: We write: = ) ( lim 2 x g x and = + ) ( lim 2 x g x and - are descriptions that tells us why the limit DOES NOT EXIST . This is not an actual limit. We would also say the line x = 2 is a vertical asymptote ....
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Overview - we get closer and closer to the y-value 4, we...

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