Visual Guide for Lab 12 In this exercise, we are examining two ideas. First, we want to understand the limit of a sequence of functions. Then we will examine the two kinds of convergence that are possible when talking about sequences of functions. Example 1. In this example, we consider the functions (x^n) on the common domain [0,1]. We want to understand graphically the pointwise limit of this sequence. We will graph some of the functions in this sequence and then focus on the value of the limit function at a particular point in the domain. We begin by defining the sequence of functions. We enter the formula for the first 300 functions in the sequence. The number 300 was chosen arbitrarily. We let f[n](x) = x^n. P:=x: for k from 2 to 300 do P:= P,x^k: od: for n from 1 to 300 do f[n]:=unapply(P[n],x): od: Now we plot the functions as requested in Question #1a. plot({f[1](x),f[3](x),f[5](x),f[10](x),f[15](x)},x=0..1); Here we graph some of the functions in the sequence (x^n).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 sampf:=%: We save the above graph, should we need it.In Question #1b, we want to investigate the behavior of the sequence of functions when they are evaluated at x_0 = .5. We specify that point in the next line. x0:=.5; NiM+JSN4MEckIiImISIi Now we will evaluate the first six functions in the sequence at the point x = x_0. for n from 1 to 6 do f[n](x0); od; NiMkIiImISIi NiMkIiNEISIj NiMkIiREIiEiJA== NiMkIiREJyEiJQ== NiMkIiVESiEiJg== NiMkIiZEYyIhIic= Below we plot the points (x_0,f[n](x_0)) for n = 1 to 6. Notice they all lie on the vertical line x = x_0, as expected. pointplot({seq([x0,f[n](x0)],n=1..6)}); LSUlUExPVEc2KS0lJ1BPSU5UU0c2KDckJCIiJiEiIkYpNyRGKSQiI0QhIiM3JEYpJCIkRCIhIiQ3JEYpJCIkRCchIiU3JEYpJCIlREohIiY3JEYpJCImRGMiISInLSUlRk9OVEc2JCUqSEVMVkVUSUNBRyIjNS0lJkNPTE9SRzYjJSVOT05FRy0lK1BST0pFQ1RJT05HNiMkRkRGKy0lKkxJTkVTVFlMRUc2IyIiIS0lKkdSSURTVFlMRUc2IyUsUkVDVEFOR1VMQVJHLSUsT1JJRU5UQVRJT05HNiQkIiNYRlBGWA== b:=%: We save the above graph.We would like to look at those points as part of the graphs of the functions f[n](x). We first graph the first 6 functions in the sequence. plot({seq(f[n](x),n=1..6)},x=0..1); 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c:=%: We will display the first six points in the sequence (f[n](x_0)) as points on the corresponding functions. display(b,c); 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What does the limit of those function values at x = x_0 appear to be? Repeat this process for other values of x in the domain [0,1]. Adjust the above code as needed. We can define the pointwise limit of this sequence of functions. We do that below and then plot the pointwise limit. flim := x -> piecewise(x=0,0,x>0 and x<1,0,x=1,1); NiM+JSVmbGltR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lKnBpZWNld2lzZUc2KC85JCIiIUYxMzJGMUYwMkYwIiIiRjEvRjBGNUY1RihGKEYo plot(flim(x),x=0..1,discont=true,style = point, color = khaki); Notice that style = point was chosen so that you would get an accurate graph.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 grflim:=%: Example 2. We repeat the process we performed above using the sequence of functions ((x^n)/(1+x^n)) with common domain [0,2]. We will graph some of the functions in this sequence and then focus on the value of the limit function at a particular point in the domain. We begin by defining the sequence of functions. We enter the formula for the first 300 functions in the sequence. The number 300 was chosen arbitrarily. We let g[n](x) = (x^n)/(1+x^n). P:=x/(1+x): for k from 2 to 300 do P:= P,(x^k)/(1+x^k): od: for n from 1 to 300 do g[n]:=unapply(P[n],x): od: Now we plot the functions as requested in Question #2a. plot({g[1](x),g[10](x),g[50](x),g[100](x),g[300](x)},x=0..2); Here we graph some of the functions in the sequence 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sampg:=%: In Question #2b, we want to investigate the behavior of the sequence of functions when they are evaluated at x_0 = .5. We specify that point in the next line. x0:=.5; NiM+JSN4MEckIiImISIi Now we will evaluate the first six functions in the sequence at the point x = x_0. for n from 1 to 6 do g[n](x0); od; NiMkIitMTExMTCEjNQ== NiMkIisrKysrPyEjNQ== NiMkIis2NjY2NiEjNQ== NiMkIitUSE4jKWUhIzY= NiMkIitJSUlJSSEjNg== NiMkIitROllROiEjNg== Below we plot the points (x_0,g[n](x_0)) for n = 1 to 6. Notice they all lie on the vertical line x = x_0, as expected. pointplot({seq([x0,g[n](x0)],n=1..6)}); LSUlUExPVEc2KS0lJ1BPSU5UU0c2KDckJCIiJiEiIiQiK0xMTExMISM1NyRGKSQiKysrKys/Ri43JEYpJCIrNjY2NjZGLjckRikkIitUSE4jKWUhIzY3JEYpJCIrSUlJSUlGODckRikkIitROllROkY4LSUlRk9OVEc2JCUqSEVMVkVUSUNBRyIjNS0lJkNPTE9SRzYjJSVOT05FRy0lK1BST0pFQ1RJT05HNiMkRkNGKy0lKkxJTkVTVFlMRUc2IyIiIS0lKkdSSURTVFlMRUc2IyUsUkVDVEFOR1VMQVJHLSUsT1JJRU5UQVRJT05HNiQkIiNYRk9GVw== b:=%: We save the above graph.We would like to look at those points as part of the graphs of the functions g[n](x). We first graph the first 6 functions in the sequence. plot({seq(g[n](x),n=1..6)},x=0..2); 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c:=%: We will display the first six points in the sequence (g[n](x_0)) as points on the corresponding functions. display(c,b); 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What does the limit of those function values at x = x_0 appear to be? Repeat this process for other values of x in the domain [0,2]. Adjust the above code as needed. We can define the pointwise limit of this sequence of functions. We do that below and then plot the pointwise limit. glim:=x->piecewise(x=0,0, x<1,0,x=1,.5,1<x and x<2,1); As before, after additional evaluations, define the pointwise limit of the sequence of functions.NiM+JSVnbGltR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lKnBpZWNld2lzZUc2Ki85JCIiIUYxMkYwIiIiRjEvRjBGMyQiIiYhIiIzMkYzRjAyRjAiIiNGM0YoRihGKA== glim(1); Evaluate the limit function at the interesting point x = 1.NiMkIiImISIi Now we plot the limit function. pt1:=plot(glim(x),x=0..1,style=point, color = khaki): pt2:=plot(glim(x),x=1..2,style=point, color=khaki): display(pt1,pt2); 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 grglim:=%: Example 3. We repeat the process we performed above using the sequence of functions ((x)/(1+nx^2)) with common domain [0,1]. We will graph some of the functions in this sequence and then focus on the value of the limit function at a particular point in the domain. We begin by defining the sequence of functions. We enter the formula for the first 300 functions in the sequence. The number 300 was chosen arbitrarily. We let h[h](x) = (x)/(1+nx^2). P:=x/(1+x): for k from 2 to 300 do P:= P,(x)/(1+k*x^2): od: for n from 1 to 300 do h[n]:=unapply(P[n],x): od: Now we plot the functions as requested in Question #3a. plot({h[1](x),h[10](x),h[50](x),h[100](x),h[300](x)},x=0..1); Here we graph some of the functions in the sequence 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samph:=%: In Question #3b, we want to investigate the behavior of the sequence of functions when they are evaluated at x_0 = .5. We specify that point in the next line. x0:=.5; NiM+JSN4MEckIiImISIi Now we will evaluate the first twenty functions in the sequence at the point x = x_0. for n from 1 to 20 do h[n](x0); od; NiMkIitMTExMTCEjNQ== NiMkIitMTExMTCEjNQ== NiMkIitkRzlkRyEjNQ== NiMkIisrKysrRCEjNQ== NiMkIitBQUFBQSEjNQ== NiMkIisrKysrPyEjNQ== NiMkIis9PT09PSEjNQ== NiMkIitubW1tOyEjNQ== NiMkIitROllROiEjNQ== NiMkIitIOWRHOSEjNQ== NiMkIitMTExMOCEjNQ== NiMkIisrKytdNyEjNQ== NiMkIispZXFrPCIhIzU= NiMkIis2NjY2NiEjNQ== NiMkIit6OmpfNSEjNQ== NiMkIisrKysrNSEjNQ== NiMkIitDJjRRXyohIzY= NiMkIisiNDQ0NCohIzY= NiMkIit1QGwmcCkhIzY= NiMkIitMTExMJCkhIzY= Below we plot the points (x_0,h[n](x_0)) for n = 1 to 20. Notice they all lie on the vertical line x = x_0, as expected. pointplot({seq([x0,h[n](x0)],n=1..20)}); 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 b:=%: We save the above graph.We would like to look at those points as part of the graphs of the functions h[n](x). We first graph the first 20 functions in the sequence. plot({seq(h[n](x),n=1..20)},x=0..1); 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c:=%: We will display the first 20 points in the sequence (h[n](x_0)) as points on the corresponding functions. display(c,b); 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What does the limit of those function values at x = x_0 appear to be? Repeat this process for other values of x in the domain [0,1]. Adjust the above code as needed. We can define the pointwise limit of this sequence of functions. We do that below and then plot the pointwise limit. hlim:=x->0; As before, after additional evaluations, define the pointwise limit of the sequence of functions.NiM+JSVobGltRyIiIQ== Now we plot the limit function. plot(hlim(x),x=0..1,style = point, color=khaki); 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 grhlim:=%: Example 1 revisited. Now that we can compute the pointwise limit of a sequence of functions, we want to examine the convergence from a more global view. Do the sequences of function values converge rather uniformly, regardless of the domain values, or does the rate of convergence seem to depend upon the domain value? We will form an epsilon band about the limit function. We let "ep" represent epsilon. In the first case, epsilon is .5. ep:=.5; NiM+JSNlcEckIiImISIi Since we have already saved the graph from Question #1a of Section 2 of the lab as sampf and the graph of the limit function as grflim, we need only graph the functions f + epsilon and f- epsilon. Then we will display all the graphs. plot({flim(x)+ep,flim(x)-ep},x=0..1,style=point, color= brown); Look at how the terms of the first sequence of functions approach the limit function. Remember the limit function sits on the x-axis except at the point x=1, where the value is 1. Is there a point in the indexing set for the sequence of functions, after which the sequence functions fall entirely within the epsilon band about the limit function?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 fband:=%: display(sampf, grflim, fband); 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To experiment with graphing different functions f[n](x) in the epsilon band of the limit function, use the code below. plot(f[40](x),x=0..1); d:=%: display(d,grflim, fband); 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 Example 2 revisited. We consider the same questions for the second sequence of functions. We will form an epsilon band about the limit function. We let "ep" represent epsilon. In the first case, epsilon is .3. ep:=.3; NiM+JSNlcEckIiIkISIi Since we want to adjust the domain interval we will graph again the functions from Question #2 in Section 2 of the lab as as well as the pointwise limit. Then we will display all the graphs. plot({g[1](x),g[10](x),g[50](x),g[100](x),g[300](x)},x=.95..1.05); 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sampg2:=%: part1:=plot(glim(x),x=.95..1,style=point, color = khaki): part2:=plot(glim(x),x=1..1.05,style=point, color = khaki): display(part1,part2): glim2:=%: plot({glim(x)+ep,glim(x)-ep},x=.95..1.05,style=point, color= brown): gband:=%: display(sampg2, gband,glim2); 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To experiment with graphing different functions g[n](x) in the epsilon band of the limit function, use the code below. plot(g[40](x),x=.95..1.05); d:=%: display(d, gband,glim2); 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 Example 3 revisited. We consider the same questions for the third sequence of functions. We will form an epsilon band about the limit function. We let "ep" represent epsilon. In the first case, epsilon is .3. ep:=.3; NiM+JSNlcEckIiIkISIi Since we have already saved the graph from Question #3a of Section 2 of the lab as samph and the graph of the limit function as grhlim, we need only graph the functions h + epsilon and h- epsilon. Then we will display all the graphs. plot({hlim(x)+ep,hlim(x)-ep},x=0..1,style=point, color= brown); Look at how the terms of the first sequence of functions approach the limit function. Remember the limit function sits on the x-axis except at the point x=1, where the value is 1. Is there a point in the indexing set for the sequence of functions, after which the sequence functions fall entirely within the epsilon band about the limit function?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 hband:=%: display(samph, grhlim, hband); 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To experiment with graphing different functions h[n](x) in the epsilon band of the limit function, use the code below. plot(h[40](x),x=0..1); d:=%: display(d,grhlim, hband); 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JCEyLysrKysrKzckRlxbbSQiMi8rKysrKys3JEZcW21GZ1ttLUZfYGw2IyUlTk9ORUctJStQUk9KRUNUSU9ORzYjRmJgbC0lKkxJTkVTVFlMRUc2I0YrLSUqR1JJRFNUWUxFRzYjJSxSRUNUQU5HVUxBUkctJSxPUklFTlRBVElPTkc2JCQiI1hGK0ZlXW0= Example 4. We consider one more sequence of functions, which exhibits the same nice behavior you observed for Example 3. The sequence of functions is (n*x*(exp((-n^2)*x)) on the domain [0.1]. plot({x*exp(-x),2*x*exp(-4*x),3*x*exp(-9*x),4*x*exp(-16*x),5*x*exp(-25*x),6*x*exp(-36*x),10*x*exp(-100*x),11*x*exp(-121*x),12*x*exp(-144*x)},x=0..1); Plot some terms in the sequence and try to determine the limit 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sampk:=%: klim:=x->0; Define the limit function.NiM+JSVrbGltRyIiIQ== plot({klim(x),klim(x)+.1,klim(x)-.1},x=0..1); e:=%: display(e,sampk); Graph some of the terms of the sequence of functions, the limit function, and the epsilon 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