Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. ######################################################################## # MY_MATH_LAB_LINK # # www.coursecompass.com # # # # course id is: hentzel26905 # # # # The My Math Lab Code is bound with your book and you should # # have it already. It is valid for five semesters after you # # activate the book the first time. # ######################################################################## Friday, February 29 8.2 Other indeterminate Forms p432:12,24,41,44 / / Main Idea: More techniques to use with L Hopital s Rule ln(x) P(x) Key Words: ------- --------- P(x) x e Goal: An education provides the information to answer a question. Not the answer to the question. -------------------------------------------------------- Previous assignment: p427: (14),(16),(22),(26) Page 427 Problem 14 3 Sin[x] 3 Cos[x] Lim ----------- = Lim ------------- = Lim -6 Cos[x] Sqrt[-x] = 0 x->0- Sqrt[-x] x->0- -1/2 x->0 -1/2 (-x) ------------------------------------------------------------------------- Get["font.math"]; P1 = Plot[(3 Sin[x])/Sqrt[-x],{x,-Pi,0},AspectRatio->Automatic,PlotPoints->100]; P2 = Plot[-6 Cos[x] Sqrt[-x],{x,-Pi/2,0},AspectRatio->Automatic, PlotStyle->{RGBColor[1,0,0]},PlotPoints->100]; P3 = ListPlot[ {{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P4 = Show[P1,P2,P3,PlotLabel->"P 427 p14, black = 3 Sin[x]/Sqrt[-x] red=f'/g' ", AspectRatio->Automatic]; Display["14x.ps",P4]; P5 = Plot[(3 Sin[x])/Sqrt[-x],{x,-0.04,0},AspectRatio->Automatic, PlotPoints->100,PlotStyle->{RGBColor[0,0,1],Thickness[0.03]}, PlotRange->All]; P6 = Plot[-6 Cos[x] Sqrt[-x], {x,-0.01,0},AspectRatio->Automatic, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]},PlotPoints->100]; P7 = ListPlot[ {{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.1 ]}]; P8 = Show[P5,P6,P7,PlotLabel->"P 427 p14, blue = 3 Sin[x]/Sqrt[-x] red=f'/g' ", Axes->False] Display["14y.ps",P8]; ------------------------------------------------------------------------- Page 427 Problem 16 2 Sin[x]-Tan[x] Cos[x] - Sec [x] Lim ------------- = Lim ---------------------- x->0 2 x->0 2 x Sin[x] 2x Sin[x] + x Cos[x] 2 -Sin[x]-2 Sec [x] Tan[x] = Lim -------------------------------------------- x->0 2 2 Sin[x] + 2 x Cos[x] + 2x Cos[x]-x Sin[x] 2 -Sin[x] - 2 Sec [x] Tan[x] = Lim -------------------------------------------- x->0 2 2 Sin[x] + 4 x Cos[x] -x Sin[x] 2 2 4 -Cos[x] -4 Sec [x] Tan [x] -2 Sec [x] = Lim -------------------------------------------------------- x->0 2 2 Cos[x] + 4 Cos[x] -4 x Sin[x] -2 x Sin[x] - x Cos[x] 2 2 4 -Cos[x] -4 Sec [x] Tan [x] -2 Sec [x] = Lim ------------- ------------------------------------------ x->0 2 6 Cos[x] -6 x Sin[x] - x Cos[x] -3 = -------- = -1/2 <==== Answer 6 ----------------------------------------------- Get["font.math"]; f[x_] = Sin[x]-Tan[x]; g[x_] = x^2 Sin[x]; fp[x_] = D[f[x],x]; gp[x_] = D[g[x],x]; fpp[x_] = D[fp[x],x]; gpp[x_] = D[gp[x],x]; fppp[x_] = D[fpp[x],x]; gppp[x_] = D[gpp[x],x]; P1 = Plot[f[x]/g[x],{x,-Pi/4,Pi/4},PlotStyle->{RGBColor[0,0,0]}]; P2 = Plot[fp[x]/gp[x],{x,-Pi/4,Pi/4},PlotStyle->{RGBColor[1,0,0]}]; P3 = Plot[fpp[x]/gpp[x],{x,-Pi/4,Pi/4},PlotStyle->{RGBColor[0,1,0]}]; P4 = Plot[fppp[x]/gppp[x],{x,-Pi/8,Pi/8},PlotStyle->{RGBColor[0,0,1]}];; P5 = Plot[ 0,{x,-Pi/4, Pi/4},PlotStyle->{Thickness[0.005]}]; P6 = Show[P1,P2,P3,P4,P5,AspectRatio->Automatic, PlotRange->{{-Pi/4,Pi/4},{-1,0}}, PlotLabel->"P427 p16 black = (Sin[x]-Tan[x])/(x^2 Sin[x],'/', ''/'','''/'''"]; Display["16x.ps",P6]; P7 = Plot[ f[x],{x,-0.2 ,0.2 },PlotStyle->{RGBColor[1,0,0]}]; P8 = Plot[ g[x],{x,-0.2 ,0.2 },PlotStyle->{RGBColor[0,1,0]}]; P9 = Show[P7,P8,PlotLabel->"P427 p16,red = (Sin[x]-Tan[x]);green = x^2 Sin[x]"]; Display["16y.ps",P9]; ----------------------------------------------------- Page 427 Problem 22 Lim Sin[x] + Tan[x] x->0- ------------------ x -x e + e - 2 2 Lim Cos[x] + Sec [x] x->0- ------------------ = - Infinity x -x e - e ------------------------------------------------------------ Get["font.math"]; f[x_] := Sin[x] + Tan[x]; fp[x_] = D[f[x],x]; g[x_] := E^x + E^(-x) -2; gp[x_] = D[g[x],x]; P1 = Plot[f[x]/g[x],{x,-1.00,-0.01}]; P2 = Plot[f[x]/g[x],{x,+0.01,+1.00}]; P3 = Plot[fp[x]/gp[x],{x, -1,-0.01},PlotStyle->{RGBColor[1,0,0]}]; P4 = Plot[fp[x]/gp[x],{x,+0.01,+1.00},PlotStyle->{RGBColor[1,0,0]}]; P5 = Show[P1,P2,P3,P4, PlotLabel->"P427 p22 black=(Sin[x]-Tan[x])/(e^x+x^(-x)-2, red='/'"]; Display["22x.ps",P5]; P6 = Plot[ f[x],{x,-0.4 ,0.4 },PlotStyle->{RGBColor[1,0,0]}]; P7 = Plot[ g[x],{x,-0.4 ,0.4 },PlotStyle->{RGBColor[0,1,0]}]; P8 = Show[P6,P7, PlotLabel->"P427 p22,red = (Sin[x]+Tan[x]);green = e^x+e^(-x)-2"]; Display["22y.ps",P8]; ------------------------------------------------------------ Page 427 Problem 26 2 2 2 x Sin[1/x] 2x Sin[1/x] + x Cos[1/x] (-1/x ) Lim ------------- = Lim --------------------------------- = x->0 Tan[x] x->0 2 Sec [x] 2x Sin[1/x] - Cos[1/x] = Lim ----------------------- x->0 2 Sec [x] / The limit does not exist and it is not 0/0 so we cannot use L Hopitals rule again. But this does NOT say the original function does not have a limit. _ _ _ _ | 2 | | | | x Sin[1/x] | | x | |---------------- | = |--------|( x ) ( Sin[1/x] Cos[x] ) = 0 | Tan[x] | | Sin[x] | |_ _| |_ _| -------------------------------------------------------- Get["font.math"]; f[x_] := x^2 Sin[1/x]; g[x_] := Tan[x]; fp[x_] = D[f[x],x]; gp[x_] = D[g[x],x]; P1 = Plot[f[x]/g[x],{x,-Pi/2,Pi/2},PlotStyle->{RGBColor[1,0,0]}]; P2 = Plot[fp[x]/gp[x],{x,-Pi/2,Pi/2},PlotStyle->{RGBColor[0,1,0]}]; P3 = Plot[{x,-x},{x,-Pi/4,Pi/4},PlotStyle->{RGBColor[0,0,1]}]; P4 = Show[P3,P2,P1,PlotLabel->"P427 p26; red=x^2 Sin[x]/Tan[x],green='/'", AspectRatio->Automatic]; Display["26.ps",P4]; --------------------------------------------------------------- New Material: x Page 429 Example 1. Find Lim ------------ x->Infinity x e x 1 Lim --------------- = Lim -------------- = 0 x->Infinity x x->Infinity x e e p(x) Example: Show that Lim -------------- = 0 x->Infinity x e where p(x) is any polynomial. / x Applying L Hopitals rule several times, the denominator remains e and the numerator will be eventually reduced to a constant. Thus the limit is zero. a x Page 420 Example 2. Show that Lim ----------- = 0 x->Infinity x e Using L'Hopital's rule several times, leaves a negative power of x in the numerator. The denominator is still x e so the limit is zero. ln(x) Page 429 Example 3: Show Lim ----------- = 0 ( a > 0 ) x->Infinity a x ln(x) 1/x 1 Lim -------- = Lim ---------- = --------- = 0 x->Infinity a a-1 a x a x a x Page 430 Example 4 ln(x) 1/x Sin[x] Sin[x] Lim -------- = --------- = - ------ ------- = 0 + Ctn(x) 2 x 1 x->0 -Csc [x] ----------------------------------------- Get["font.math"]; P1 = Plot[Log[x],{x,0.000,0.002},PlotStyle->{RGBColor[1,0,0]},PlotPoints->100]; P2 = Plot[Cot[x],{x,0.001,0.002},PlotPoints->100,PlotStyle->{RGBColor[0,1,0]}]; P3 = Show[P1,P2,PlotLabel->"P430 Ex 4 red= ln[x];green=Ctn[x]"]; Display["ex4x.ps",P3]; P4 = Plot[ Log[x]/Cot[x],{x,0,1},PlotPoints->100,AspectRatio->Automatic]; P5 = ListPlot[{{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P6 = Show[P4,P5,PlotLabel->"P 430 Ex4 ln[x]/Ctn[x]",AspectRatio->Automatic]; Display["ex4y.ps",P6]; ----------------------------------- Page 430 Example 5 Lim Tan[x] ln(Sin[x]) x-> Pi/2 ----------------------------------------------- Get["font.math"]; P1 = Plot[Tan[x] Log[Sin[x]],{x,0, Pi}]; P2 = Plot[-Cos[x] Sin[x],{x,0, Pi},PlotStyle->{RGBColor[1,0,0]}]; P3 = ListPlot[{{0,0}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P4 = Show[P1,P2,P3, PlotLabel->"P 430 Ex5; Tan[x] Log[ Sin[x]] at Pi/2; red='/' "]; Display["ex5.ps",P4]; ---------------------------------------------- ln(Sin[x]) Lim Tan[x] ln(Sin[x]) = Lim ---------- x->Pi/2 x->Pi/2 Ctn[x] Cos[x] ------- Sin[x] = -------------- = Lim -Cos[x] Sin[x] = 0 2 x->Pi/2 - Csc [x] ----------------------------------------------------- Page 431 Example 6 x 1 Lim --------- - -------- + x-1 ln(x) x->1 x 1 x ln(x) -x+1 Lim --------- - -------- = lim ---------------- + x-1 ln(x) + (x-1) ln(x) x->1 x->1 ln(x) + 1 -1 x ln(x) = lim ------------- = lim -------------- + ln(x) +(x-1)/x + x ln(x) + x-1 x->1 x->1 ln(x) + 1 = lim --------------- = 1/2 + ln(x)+1+1 x->1 ---------------------------------------------------------- Get["font.math"]; P1 = Plot[ x/(x-1),{x,0, 0.9},PlotPoints->100,PlotStyle->{RGBColor[1,0,0]}]; P2 = Plot[ x/(x-1),{x,1.1, 2.0},PlotPoints->100,PlotStyle->{RGBColor[1,0,0]}]; P3 = Plot[1/Log[x],{x,1.1, 2.0},PlotPoints->100,PlotStyle->{RGBColor[0,1,0]}]; P4 = ParametricPlot[{1,y},{y,-10,10},PlotStyle->{RGBColor[0,0,1]}]; P5 = Show[P1,P2,P3,P4,PlotLabel->"P411 Ex6 red=x/x-1; green=1/ln(x)"]; Display["ex6x.ps",P5]; P6 = Plot[x/(x-1) - 1/Log[x],{x,0,2}]; P7 = ListPlot[{{1,1/2}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P8 = Show[P6,P7,PlotLabel->"P 431 Ex6 x/(x-1)-1/Log[x]",AspectRatio->Automatic]; Display["ex6y.ps",P8]; ---------------------------------------------------------- Page 431 Example 7 Ctn[x] Lim (x+1) + x->0 ln(x+1) Lim ln(y) = Lim Ctn[x] ln(x+1) = Lim -------- + + + Tan[x] x->0 x->0 x->0 1/(x+1) = Lim ------------- = 1 + 2 x->0 Sec [x] Lim y = e. + x->0 ----------------------------------------------------- Get["font.math"]; P1 = Plot[ (x+1)^Cot[x],{x,-0.8, 3.15 }]; P2 = ListPlot[{{0,E}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P3 = Plot[0,{x,-0.8, 1}]; P4 = Show[P1,P2,P3,PlotLabel->"P431 Ex7 (x+1)^Ctn[x] ", PlotRange->{{-0.8,3.15},{0,3}}]; Display["ex7.ps",P4]; ----------------------------------------------------- Page 431 Example 8 Cos[x] Lim (Tan[x]) ___ x-> Pi/2 ln(Tan[x]) ln(y) = Lim Cos[x] ln(Tan[x]) = Lim ---------- ___ ___ Sec[x] x->Pi/2 x->Pi/2 2 1/Tan[x] Sec [x] Sec[x] = Lim ------------- = Lim ----------- ___ Sec[x] Tan[x] ___ 2 x->Pi/2 x->Pi/2 Tan [x] 2 Cos [x] Cos[x] = Lim ---------------- = Lim --------- = 0 ___ 2 ___ 2 x->Pi/2 Cos[x] Sin [x] x->Pi/2 Sin [x] 0 y = e = 1 ----------------------------------------------------- Get["font.math"]; P1 = Plot[ Tan[x]^Cos[x],{x,0,Pi/2}]; P2 = ListPlot[{{Pi/2,1}},PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}]; P3 = Show[P1,P2,PlotLabel->"P431 Ex 8; Tan[x]^Cos[x] at Pi/2"]; Display["ex8.ps",P3]; -----------------------------------------------------