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Foundations of Calculus
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understanding of limit We now develop that understanding with some carefully chosen examples EXAMPLE 21
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De ne f (x) = 3 x x +1
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See Fig 21 Calculate limx 1 f (x)
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SOLUTION Observe that, when x is to the left of 1 and very near to 1 then f (x) = 3 x is very near to 2 Likewise, when x is to the right of 1 and very near to 1 then f (x) = x 2 + 1 is very near to 2 We conclude that
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We have successfully calculated our rst limit Figure 21 con rms the conclusion that our calculations derived EXAMPLE 22
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De ne g(x) = Calculate limx 2 g(x) x2 4 x 2
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SOLUTION We observe that both the numerator and the denominator of the fraction de ning g tend to 0 as x 2 (ie, as x tends to 2) Thus the question seems to be indeterminate However, we may factor the numerator as x 2 4 = (x 2)(x + 2) As long as x = 2 (and these are the only x that we examine when we
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AM FL Y
Fig 21
lim f (x) = 2
CHAPTER 2 Foundations of Calculus
calculate limx 2 ), we can then divide the denominator of the expression de ning g into the numerator Thus g(x) = x + 2 Now
for x = 2
lim g(x) = lim x + 2 = 4
Fig 22
The graph of the function g is shown in Fig 22 We encourage the reader to use a pocket calculator to calculate values of g for x near 2 but unequal to 2 to check the validity of our answer For example, x 18 19 199 1999 2001 201 21 22 g(x) = [x 2 4]/[x 2] 38 39 399 3999 4001 401 41 42
We see that, when x is close to 2 (but unequal to 2), then g(x) is close (indeed, as close as we please) to 4 You Try It: Calculate the limit limx 3 x 3 3x 2 + x 3 x 3
Math Note: It must be stressed that, when we calculate limx c f (x), we do not evaluate f at c In the last example it would have been impossible to do so We want to determine what we anticipate f will do as x approaches c, not what value (if any) f actually takes at c The next example illustrates this point rather dramatically
Foundations of Calculus
Fig 23
EXAMPLE 23
De ne h(x) = Calculate limx 7 h(x) 3 1 if x = 7 if x = 7
SOLUTION It would be incorrect to simply plug the value 7 into the function h and thereby to conclude that the limit is 1 In fact when x is near to 7 but unequal to 7, we see that h takes the value 3 This statement is true no matter how close x is to 7 We conclude that limx 7 h(x) = 3 You Try It: Calculate limx 4 [x 2 x 12]/[x 4]
ONE-SIDED LIMITS
x c
There is also a concept of one-sided limit We say that lim f (x) =
if the values of f become closer and closer to when x is near to c but on the left In other words, in studying limx c f (x), we only consider values of x that are less than c Likewise, we say that
x c+
lim f (x) =
if the values of f become closer and closer to when x is near to c but on the right In other words, in studying limx c+ f (x), we only consider values of x that are greater than c
CHAPTER 2 Foundations of Calculus
EXAMPLE 24
Discuss the limits of the function f (x) = at c = 2 2x 4 x
if x < 2 if x 2
SOLUTION As x approaches 2 from the left, f (x) = 2x 4 approaches 0 As x approaches 2 from the right, f (x) = x 2 approaches 4 Thus we see that f has left limit 0 at c = 2, written
x 2
lim f (x) = 0,
and f has right limit 4 at c = 2, written
x 2+
lim f (x) = 4
Note that the full limit limx 2 f (x) does not exist (because the left and right limits are unequal) You Try It: Discuss one-sided limits at c = 3 for the function x 3 x if x < 3 f (x) = 24 if x = 3 4x + 1 if x > 3 All the properties of limits that will be developed in this chapter, as well as the rest of the book, apply equally well to one-sided limits as to two-sided (or standard) limits