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Method 1: lim 1 n 6 0, in fact it doesn t exist. Then by Problem 2.26 the series cannot converge, i.e., it diverges.

(a) Every polynomial function is a continuous function. This follows from example (b) of Property V in Section 8.2. (b) Every rational function h(x) = f (x) , where f and g are polynomials, is continuous at every real number a except the real roots g(x) (if any) of g(x). This follows from Property VI in Section 8.2.

Method 2: The sequence of partial sums is 1; 1 1; 1 1 1; 1 1 1 1; . . . i.e., 1; 0; 1; 0; 1; 0; 1; . . . . Since this sequence has no limit, the series diverges.

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There are certain properties of continuity that follow directly from the standard properties of limits (Section 8.2). Assume f and g are continuous at a. Then, (1) The sum f + g and the difference f g are continuous at a.

n!1 n!1

A three-element array of doubles specifying the center of the circle in the WCS The radius of the circle

v1 v2 vn 0 if lim vn 0. Now n!1 n v1 v2 vn v1 v2 vP vP 1 vp 2 vn n n n We must show that lim

notation f + g is a function such that ( f + g)(x) = f (x) + g(x) for every x that is in both the domain of f and the domain of g. Similarly, ( f g)(x) = f (x) g(x) for all x common to the domains of f and g.

v 1 v 2 v n @ jv1 v2 vP j jvP 1 j jvP 2 j jvn j n n n Then

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Thus, the slope of the tangent line at (x, 1/x) is 1/x 2 . (b) From part (a), the slope of the tangent line at (2, 1 ) is 2 1 1 1 2 = 2 = 4 x 2 Hence, the slope-intercept equation of the tangent line L has the form 1 y = x+b 4

jvP 1 j jvP 2 j jvn j =2 =2 =2 n P =2 < < n n n 2 After choosing P we can choose N so that for n > N > P, jv1 v2 vP j < n 2 Then using (2) and (3), (1) becomes v 1 v 2 v n < 2 2 n thus proving the required result.

CHAP. 11]

Figure 8-2. The Circle object The following example creates a circle based on user input: Public Sub TestAddCircle() Dim varCenter As Variant Dim dblRadius As Double Dim objEnt As AcadCircle On Error Resume Next '' get input from user With ThisDrawing.Utility varCenter = .GetPoint(, vbCr & "Pick the centerpoint: ") dblRadius = .GetDistance(varCenter, vbCr & "Enter the radius: ") End With

Now by the binomial theorem,

11.4 (a) Find a formula for the slope of the tangent line at any point of the graph of the function f such that f (x) = 3x 2 6x + 4. (b) Find the slope-intercept equation of the tangent line at the point (0, 4) of the graph. (c) Draw the graph of f and show the tangent line at (0, 4).

CHAP. 2]

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