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print barcode image c# Indeterminate Forms in Software
Indeterminate Forms Encoding QRCode In None Using Barcode drawer for Software Control to generate, create QR Code image in Software applications. Decode QR Code In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. 511 INTRODUCTION
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Drawing EAN128 In None Using Barcode generator for Software Control to generate, create EAN / UCC  13 image in Software applications. Make UCC  12 In None Using Barcode drawer for Software Control to generate, create UPCA image in Software applications. If limx c f (x) exists and limx c g(x) exists and is not zero then the limit ( ) is straightforward to evaluate However, as we saw in Theorem 23, when limx c g(x) = 0 then the situation is more complicated (especially when limx c f (x) = 0 as well) For example, if f (x) = sin x and g(x) = x then the limit of the quotient as x 0 exists and equals 1 However if f (x) = x and g(x) = x 2 then the limit of the quotient as x 0 does not exist In this section we learn a rule for evaluating indeterminate forms of the type ( ) when either limx c f (x) = limx c g(x) = 0 or limx c f (x) = limx c g(x) = Such limits, or forms, are considered indeterminate because the limit of the quotient might actually exist and be nite or might not exist; one cannot analyze such a form by elementary means Creating Code 39 In None Using Barcode generator for Software Control to generate, create Code 39 Full ASCII image in Software applications. Encode Code 128 Code Set B In None Using Barcode creation for Software Control to generate, create Code 128A image in Software applications. Copyright 2003 by The McGrawHill Companies, Inc Click Here for Terms of Use
Create Barcode In None Using Barcode generation for Software Control to generate, create bar code image in Software applications. UPC  13 Maker In None Using Barcode creator for Software Control to generate, create GS1  13 image in Software applications. 124 512 Planet Creation In None Using Barcode maker for Software Control to generate, create USPS Confirm Service Barcode image in Software applications. Bar Code Recognizer In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. CHAPTER 5 L H PITAL S RULE
Drawing Bar Code In None Using Barcode creator for Excel Control to generate, create barcode image in Excel applications. Printing UPCA Supplement 5 In ObjectiveC Using Barcode drawer for iPhone Control to generate, create UPC Symbol image in iPhone applications. Indeterminate Forms
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f (x) f (x) = lim , x c g (x) g(x) provided this last limit exists as a nite or in nite limit Let us learn how to use this new result EXAMPLE 51 Evaluate
x 1 x 2
ln x
+x 2
SOLUTION We rst notice that both the numerator and denominator have limit zero as x tends to 1 Thus the quotient is indeterminate at 1 and of the form 0/0 l H pital s Rule therefore applies and the limit equals (d/dx)(ln x) , x 1 (d/dx)(x 2 + x 2) lim provided this last limit exists The last limit is 1/x 1 = lim 2+x x 1 2x + 1 x 1 2x lim Therefore we see that x 1 x 2
1 ln x = 3 +x 2
You Try It: Apply l H pital s Rule to the limit limx 2 sin( x)/(x 2 4) You Try It: Use l H pital s Rule to evaluate limh 0 (sin h/ h) and limh 0 (cos h 1/ h) These limits are important in the theory of calculus EXAMPLE 52 Evaluate the limit
x 0 x3 x sin x
CHAPTER 5 Indeterminate Forms
SOLUTION As x 0 both numerator and denominator tend to zero, so the quotient is indeterminate at 0 of the form 0/0 Thus l H pital s Rule applies Our limit equals (d/dx)x 3 , x 0 (d/dx)(x sin x) lim provided that this last limit exists It equals 3x 2 x 0 1 cos x lim This is another indeterminate form So we must again apply l H pital s Rule The result is 6x lim x 0 sin x This is again indeterminate; another application of l H pital s Rule gives us nally 6 = 6 x 0 cos x lim We conclude that the original limit equals 6 You Try It: Apply l H pital s Rule to the limit limx 0 x/[1/ ln x] Indeterminate Forms Involving We handle indeterminate forms involving in nity as follows: Let f (x) and g(x) be differentiable functions on (a, c) (c, b) If lim f (x) and lim g(x) both exist and equal + or (they may have the same sign or different signs) then f (x) f (x) = lim , x c g(x) x c g (x) lim provided this last limit exists either as a nite or in nite limit Let us look at some examples EXAMPLE 53 Evaluate the limit
x 0 lim x 3 ln x 
SOLUTION This may be rewritten as ln x x 0 1/x 3 lim
Indeterminate Forms
Notice that the numerator tends to and the denominator tends to as x 0 Thus the quotient is indeterminate at 0 of the form / + So we may apply l H pital s Rule for in nite limits to see that the limit equals 1/x = lim x 3 /3 = 0 x 0 3x 4 x 0 Yet another version of l H pital s Rule, this time for unbounded intervals, is this: Let f and g be differentiable functions on an interval of the form [A, + ) If limx + f (x) = limx + g(x) = 0 or if limx + f (x) = and limx + g(x) = , then lim f (x) f (x) = lim x + g(x) x + g (x) lim provided that this last limit exists either as a nite or in nite limit The same result holds for f and g de ned on an interval of the form ( , B] and for the limit as x EXAMPLE 54

