# If b then we define x fblimfx and if a then we define

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If b then we define xF(b)limF(x)and if a then we define xF(a)lim F(x)If these limits exist and are finite then we say the integral _____________. Otherwise the integral ____________. For examples 1 to 13, evaluate the integral if it converges, or show that it diverges. Example 1: 11dxxThe textbook way to do Example 1: Note: The textbook expects you to use limit notation, but for me it is fine if you use the shorthand notation like in the first problem above, but make sure you understand that whenever we “plug-in” infinity we are really finding a limit.
Page 71 Example 2: 211dxxThis integral gives the volume of the solid formed by rotating the curve in Example 1 around the x-axis. Example 3: x0xedxExample 4: xxedx1e
Page 72 Example 5: 21dx1xExample 6: 21dx1x
Page 73 Improper integrals with discontinuities at or between the limits of integrationHere again we’ll do things a bit differently that the textbook. It turns out that the Fundamental Theorem is still true even if f(x)has a finite number of discontinuities on the interval [a,b], as long as the antiderivative F(x)is continuous on the interval [a,b]. Therefore, in some cases it is not necessary to take limits or split the integral even when the original function is discontinuous. Listed below are the 3 cases in which we must take action. If F(x)is continuous on (a,b]but not at xathen replace a with awhere xa)lim F(x)F(aIf F(x)is continuous on [a,b)but not at xbthen replace b with bwhere xb)lim F(x)F(bIf F(x)is continuous on [a,b]except for some c in [a,b]then bcbaacF(x)]F(x)]F(x)]If these limits exist and are finite then we say the integral converges. Otherwise the integral diverges. In the third case, the integral diverges if ______________________________________. Example 7: 201dx2xExample 8: 10ln xdxx
Page 74 Example 9: 3301dx(x1)Example 10: (you try) 8311dxxExample 11: e01dxxln x
Page 75 Example 12: 01dxx(x1)Example 13: 220sec xsecx tanx dx
Page 76 Section 9.1 Sequences: Factorials: n!n(n1)(n2) ... (3)(2)(1)for n1and 0!1Example 1: Simplify (a) 4!7!(b) (n1)!n1 !Is 2n!(2n)!? How fast does the function f(n)n!grow? List increasing functions from slowest to fastest. Which grows faster, 4nor n2? Definition: A _________________is a function whose domain is the set of positive integers. For example, na3nor n21bnDef: A sequence naconverges if ____________________ where Lis a real number. If ___________________________________________________then the sequence diverges. Do the sequences naand nbabove converge or diverge?
Page 77 Theorem: Let f(x)be a function of a real variable such that f(x)has the same values as a sequence nafor all positive integers. If xlimf(x)Lthen _____________________. This theorem allows us to use limit techniques such as RFLTand LHRto sequences.
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