lim x a f x L
P13 1.3 THE LIMIT OF A FUNCTION 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 . An alternative notation for is as which is usually read “ f ( x ) approaches L as x approaches a .” lim x a f x L ( ) f x L x a
P14 1.3 THE LIMIT OF A FUNCTION 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 .
P15 1.3 THE LIMIT OF A FUNCTION Figure 2 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 ( ) f a L lim ( ) . x a f x L
P16 1.3 Example 2 Guess the value of . SOLUTION Notice that the function f ( x ) = ( x – 1)/( x 2 – 1) is not defined when x = 1. However, that doesn’t matter—because the definition of says that we consider values of x that are close to a but not equal to a . 2 1 1 lim 1 x x x lim ( ) x a f x
P17 1.3 Example 2 SOLUTION 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 2 1 1 lim 0.5 1 x x x
P18 1.3 THE LIMIT OF A FUNCTION Example 2 is illustrated by the graph of f in Figure 3.
P19 1.3 THE LIMIT OF A FUNCTION Now, let’s change f slightly by giving it the value 2 when x = 1 and calling the resulting function g : This new function g still has the same limit as x approaches 1. See Figure 4. 2 1 if 1 1 2 if 1 x x g x x x
P20 1.3 Example 3 Estimate the value of . SOLUTION The table lists values of the function for several values of t near 0. As t approaches 0, the values of the function seem to approach 0.16666666… So, we guess that: 2 2 0 9 3 lim t t t 2 2 0 9 3 1 lim 6 t t t
P21 1.3 THE LIMIT OF A FUNCTION What would have happened if we had taken even smaller values of t ? The table shows the results from one calculator. You can see that something strange seems to be happening. If you try these calculations on your own calculator, you might get different values but, eventually, you will get the value 0 if you make t sufficiently small.
P22 1.3 THE LIMIT OF A FUNCTION Does this mean that the answer is really 0 instead of 1/6? No, the value of the limit is 1/6, as we will show in the next section. The problem is that the calculator gave false values because is very close to 3 when t is small.
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- Fall '19
- Calculus, Limit, One-sided limit