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(f ) lim (7x 3 5x 2 + 2x 4) x 2 x 12 x 4 x 4 x 4 + 3x 3 13x 2 27x + 36 (l) lim x 1 x 2 + 3x 4 4 1 (o) lim 2 x 2 x 2 x 4 (i) lim

i.e., the integral along P1 AP2 (path 1) integral along P1 BP2 (path 2), and so the integral is independent of the path joining P1 and P2 . This explains the results of Problem 16.10, since f z z2 is analytic.

(h) lim (x [x])

Fig. 16-5

This example gets a distance from the user, converts it to a double using DistanceToReal with a default of architectural, and then converts and displays the results using RealToString with a unit of architectural and a precision of four decimal places: Public Sub TestRealToString() Dim strInput As String Dim strOutput As String Dim dblDist As Double With ThisDrawing.Utility strInput = .GetString(True, vbCr & "Enter a distance: ") dblDist = .DistanceToReal(strInput, acArchitectural) .Prompt vbCr & "Double: " & dblDist strOutput = .RealToString(dblDist, acArchitectural, 4) .Prompt vbCrLf & "Distance: " & strOutput End With End Sub

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f (x + h) f (x) f (x + h) f (x) and then lim (if the latter exists) for each of the following functions: h h h 0 1 (c) f (x) = 7x + 12 (b) f (x) = (a) f (x) = 3x 2 + 5 x+1 (d) f (x) = x 3 (e) f (x) = x (f ) f (x) = 5x 2 2x + 4

16.12. If f z is analytic within and on the boundary of a region bounded by two closed curves C1 and C2 (see Fig. 16-6), prove that f z dz f z dz

8.8 Assuming that lim f (x) = L and lim g(x) = K, prove rigorously: (a) lim c f (x) = c L, where c is any real number.

As in Fig. 16-6, construct line AB (called a cross-cut) connecting any point on C2 and a point on C1 . By Cauchy s theorem (Problem 16.11), f z dz 0

f (x)

since f z is analytic within the region shaded and also on the boundary. Then f z dz f z dz f z dz f z dz 0 1

Use the AngleFromXAxis method to measure the angle, in radians, between an imaginary line formed by two points and the WCS X-axis. This method returns a double. It has the following syntax: dblAngle = UtilityObject.AngleFromXAxis(StartPoint, EndPoint) Table 7-15 explains this method s parameters. Table 7-15. The AngleFromXAxis Method s Parameters

[Hints: In part (d), for L > 0, f (x) L = ( f (x) L) f (x) + f (x) + L L = f (x) L |f (x) L| L

CHAP. 16]

In part (f ), if f (x) and h(x) lie within the interval (L , L + ), so must g(x).] 8.9 In an epsilon-delta proof of the fact that lim (2 + 5x) = 17, which of the following values of is the largest that can be used,

i.e.,

given (a) (b) 2 (c) 4 (d) 5 (e) 8

& dz 2 i if n 1 16.13. (a) Prove that , where C is a simple closed curve bounding 0 if n 2; 3; 4; . . . z a n C a region having z a as interior point. (b) What is the value of the integral if n 0; 1; 2; 3; . . .

A three-element array of doubles that specifies the first point of an imaginary line in WCS A three-element array of doubles that specifies the second point of an imaginary line in WCS

x + 21 5 (b) GC Solve part (a) with a graphing calculator by graphing y = and tracing along the curve as x approaches 4. x 4 8.12 Find the following limits: (a) lim

(a) Let C1 be a circle of radius having center at z a (see Fig. 16-7). Since z a n is analytic within and on the boundary of the region bounded by C and C1 , we have by Problem 16.12, dz dz n n C z a C1 z a

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