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.166.05 Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. February 18 9.2 Indeterminent foms. p413: 1-6, 11,(14),15,(16),19 (24),27,39,(45) Main Idea: More techniques to use with L'Hopital's Rule Key Words: Goal: -------------------------------------------------------- Previous assignment: February 16 9.1 Type 0/0 p407:1-3,(4), 9-11,(12),19,20,23,(26),(30) Page 407 Problem 4 3 -------- ArcTan[3x] 1+9x^2 3 Lim ------------ = Lim --------- = ----------- = 3 x-> 0 ArcSin[x] x->0 1 1/1 ----------- Sqrt[1-x^2] -------------------------------------------------------- f[x_] := ArcTan[3x]; g[x_] := ArcSin[x]; a = Plot[{f[x],g[x]},{x,0,1}]; b = Plot[3ArcSin[x],{x,0,1},PlotStyle->RGBColor[1,0,0]]; c = Show[a,b,PlotLabel->"Page 407 Problem 4",AspectRatio->Automatic]; Display["4.ps",c]; -------------------------------------------------------- Page 407 Problem 12 7^Sqrt[x] - 1 7^Sqrt[x] 1/2 x^(-1/2) ln[7] Lim ---------------- = Lim ----------------------------------- x->0+ 2^Sqrt[x] - 1 x->0+ 2^Sqrt[x] 1/2 x^(-1/2) ln[2] 7^Sqrt[x] ln[7] Lim ---------------- = ln[7]/ln[2] x->0+ 2^Sqrt[x] ln[2] -------------------------------------------------------- f[x_] := 7^Sqrt[x]-1; g[x_] := 2^Sqrt[x]-1; a = Plot[{f[x],g[x]},{x,0,1}]; b = Plot[Log[7]/Log[2] g[x],{x,0,1},PlotStyle->RGBColor[1,0,0]]; c = Show[a,b,PlotLabel->"Page 407 Problem 12",AspectRatio->Automatic]; Display["12.ps",c]; a = Plot[f[x],{x,0,3}]; b = Plot[g[x],{x,0,16}]; c = Plot[Log[7]/Log[2] g[x],{x,0,10},PlotStyle->RGBColor[1,0,0]]; d = Show[a,b,c,PlotLabel->"Page 407 Problem 12",AspectRatio->Automatic]; Display["12x.ps",d]; -------------------------------------------------------- Page 407 Problem 26 2 x Sin[1/x] Find Lim ------------- x->0 Tan[x] -------------------------------------------------------- f[x_] := x^2 Sin[1/x]; g[x_] := Tan[x]; a = Plot[f[x]/g[x],{x,0,0.01},PlotStyle->{RGBColor[1,0,0]}]; b = Show[a,PlotLabel->"Page 407 Problem 26"; Display["26.ps",b]; -------------------------------------------------------- 2 2 | x Sin[1/x] | | x | | x | | ------------ | < | ------- | = -------- | x | | Cos[x] | = 0 | Tan[x] | | Tan[x] | |Sin[x]| 2 x Sin[1/x] Thus Lim ------------- = 0. x->0 Tan[x] ---------------------------------------------------- Page 407 Problem 30 Determine constants a,b, and c so that 4 3 ax + bx + 1 Lim ------------------- = c a+b = -1 x->1 (x-1) Sin[ Pi x ] 3 2 4 a x + 3 b x Lim -------------------------------------- 4a+3b = 0 x->1 Sin[Pi x] + (x-1) Pi Cos[ Pi x] 2 12 a x + 6 b x Lim ---------------------------------------------------------- x->1 2 Pi Cos[ Pi x] + Pi Cos[Pi x] + (x-1) Pi (-Sin[Pi x]) 12 a + 6 b ----------------------- = c -2 Pi a+b = -1 4a+3b = 0 a = 3 b = -4 36 - 24 -6 ------------ = ----- = c -2 Pi Pi -------------------------------------------------------- Limit[ (a x^4 + b x^3 + 1)/ ((x-1) Sin[ Pi x]),{x->1}] Limit[ (3 x^4 - 4 x^3 + 1)/ ((x-1) Sin[ Pi x]),{x->1}] f[x_] := 3x^4 - 4 x^3 + 1; g[x_] := (x-1) Sin[ Pi x]; a = Plot[f[x]/g[x],{x,0.9,1.1}]; b = Plot[-6/Pi,{x,0.9,1.1},PlotStyle->{RGBColor[1,0,0]}]; c = Show[a,b,PlotLabel->"Page 407 Problem 30"]; Display["30.ps",c]; h[a_,b_,x_] := (a x^4 + b x^3 + 1)/ ((x-1) Sin[ Pi x]); H = Table[ Plot[h[a,b,x],{x,0.5,1.5}],{a,1,6},{b,-6,0}]; a = Plot[h[3,-4,x],{x,0.5,1.5},PlotStyle->{RGBColor[1,0,0]}]; b = Show[H,a,PlotLabel->"1<=a<=6,-6<=b<=0,Page 407 Problem 30"]; Display["30x.ps",b]; ================================================================ New Material: x Example 1. Find Lim ------------ x->Infinity x e x 1 Lim --------------- = Lim -------------- = 0 x->Infinity x x->Infinity x e e p(x) Example 2. 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 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) Show Lim ----------- = 0 x->Infinity a x ln(x) 1/x 1 Lim -------- = Lim ---------- = --------- = 0 x->Infinity a a-1 a x a x a x -------------------------------------------------------- Example 4 ln(x) 1/x Sin[x] Sin[x] Lim -------- = --------- = - ------ ------- = 0 + Ctn(x) 2 x 1 x->0 -Csc [x] ----------------------------------------- a = Plot[ Log[x],{x,0,1}]; b = Plot[ Ctn[x],{x,0,1}]; c = Show[a,b,PlotLabel->"Page 410 Example 4"]; Display["ex4.ps",c]; ----------------------------------- Page 407 Problem 26 f[x_] := x^2 Sin[1/x]; g[x_] := Tan[x]; fp[x_] = D[f[x],x]; gp[x_] = D[g[x],x]; a = Plot[f[x]/g[x],{x,-Pi/2,Pi/2},PlotStyle->{RGBColor[1,0,0]}]; b = Plot[fp[x]/gp[x],{x,-Pi/2,Pi/2},PlotStyle->{RGBColor[0,1,0]}]; c = Plot[{x,-x},{x,-Pi/2,Pi/2},PlotStyle->{RGBColor[0,0,1]}]; d = Show[c,b,a,PlotLabel->"Page 407 Problem 26"]; Display["ex26.ps",d]; --------------------------------------------------------------- Example 5 Lim Tan[x] ln(Sin[x]) x-> Pi/2 ----------------------------------------------- a = Plot[Tan[x] Log[Sin[x]],{x,0,2 Pi}]; b = Show[a,PlotLabel->"Page 411 Example 5"]; Display["ex5.ps",b]; ---------------------------------------------- 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] ----------------------------------------------------- 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 ---------------------------------------------------------- a = Plot[ x/(x-1),{x,0,1},PlotStyle->{RGBColor[1,0,0]}]; b = Plot[1/Log[x],{x,0,1},PlotStyle->{RGBColor[0,1,0]}]; c = Show[a,b,PlotLabel->"Page 411 Example 6"]; Display["ex6.ps",c]; ---------------------------------------------------------- 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 ----------------------------------------------------- a = Plot[ (x+1)^Cot[x],{x,-2,2}]; b = Show[a,PlotLabel->"Page 411 Example 7"]; Display["7.ps",b]; ----------------------------------------------------- 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] x->Pi/2 2 x->Pi/2 Tan [x] 2 Cos [x] Cos[x] = Lim ---------------- = Lim --------- = 0 x->Pi/2 2 x->Pi/2 2 Cos[x] Sin [x] Sin [x] 0 y = e = 1 ----------------------------------------------------- a = Plot[ Tan[x]^Cos[x],{x,0,Pi}]; b = Show[a,PlotLabel->"Page 412 Example 8"]; Display["ex8.ps",b]; -----------------------------------------------------