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# Application of Squeeze Theorem

Posted 10 years ago
 Hi - i'm having trouble understanding Mathematica point blank. I have the second edition manual but cannot find what page to reference completing this problem. What are the beginning steps I take to plug this into Mathematica? Am I writing the input wrong? I'm trying to understand how to put things in but I can't seem to figure it out. Any help or suggestions would be great - thank you.Define the function f(x)=x sin (1/x) in mathematicaa) Plot the graphs of y=f(x) together with the graphs of y= |x| and y=-|x| on the same set of axes. This will visually show that f(x) is squeezed between the latter two functions near x=0. b) Analyzing the graph obtained in part (a) near x = 0 and realizing that f(x) is squeezed between the other two functions, your intuiton should tell you that lim f(x), x, 0 must be 0 since both lim |x|, x, 0 and lim -|x|, x, 0 are clearly zero. Verify your intuition by using Mathematica to compute lim f(x), x, 0I think this is what the graph is supposed to look like - but how do I put this all in Mathematica? 5 Replies
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Posted 10 years ago
Posted 10 years ago
 Abs is the absolute value function, that is |x|.
Posted 10 years ago
 There is also a very nice related Demonstration with very short code: Squeeze Theorem Manipulate[  Plot[{x^n Sin[1/x], Abs[x^n], -Abs[x^n]}, {x, -10^-i, 10^-i},    PlotStyle -> {Black, {Thick, Red}, {Thick, Blue}},    MaxRecursion -> 8, ImagePadding -> {{10, 10}, {10, 50}},    PlotLabel ->     TraditionalForm[     Row[{Style[-Abs[x^n], Blue], " \[LessEqual] ", x^n Sin[1/x],        " \[LessEqual] ", Style[Abs[x^n], Red]}]],    ImageSize -> {550, 400}], {{n, 1}, {0, 1, 2, 3}}, {{i, 0, "zoom"}, 0, 3}] Posted 10 years ago
 I'm sorry - can you explain to me what the Abs function is used for in Mathematica? I was not putting it in the original input I had - perhaps that was the problem? Thank you so much.
Posted 10 years ago
 First try making a plot for reach of the three functions:Plot[Abs[x], {x, -0.5, 0.5}, PlotStyle -> Red]Plot[-Abs[x], {x, -0.5, 0.5}, PlotStyle -> Blue]Plot[x Sin[1/x], {x, -0.5, 0.5}, PlotStyle -> Green]You can put these all together in different ways. For example you can use Show. Additionally you can use Plot with multiple functions:Plot[{Abs[x], -Abs[x], x Sin[1/x]}, {x, -0.5, 0.5}, PlotStyle -> {Red, Blue, Green}]