Plotting and comparing functions

Often we want to plot two functions and color them differently.  The cell below is and executable cell. You execute it by pressing Ctrl+Enter; make sure that the cursor is in the cell. Change one the functions and see what happens.

Plot[{x, 2 x}, {x, -3, 3}, 
      PlotStyle -> {Red, Blue}]

We can compare two functions.

Plot[{Sin[x]/(x (((Sin[x]))^2) + 1), (1/x)}, {x, 1, 3}, 
      PlotStyle -> {Red, Blue}]

This is somewhat similar to the second problem on the second assignment.

\(Plot[{Sin[2  x]\^2\/\(x\ \((Sin[x])\)\^2 + 1\), 1\/x}, {x, 1, 30}, 
      PlotStyle -> {Red, Blue}]\)

Try a different blue

\(Plot[{Sin[2  x]\^2\/\(x\ \((Sin[x])\)\^2 + 1\), 4\/x}, {x, 1, 30}, 
      PlotStyle -> {Red, Blue}]\)

Try a larger domain

\(Plot[{Sin[2  x]\^2\/\(x\ \((Sin[x])\)\^2 + 1\), 4\/x}, {x, 200, 300}, 
      PlotStyle -> {Red, Blue}]\)

Try even larger domain

\(Plot[{Sin[2  x]\^2\/\(x\ \((Sin[x])\)\^2 + 1\), 4\/x}, {x, 2900, 3000}, 
      PlotStyle -> {Red, Blue}]\)

This strongly indicates, BUT DOES NOT PROVE the inequality   \(TraditionalForm\`Sin[2  x]\^2\/\(\(x(Sin[x])\)\^2 + 1\) < 4\/x\). This can be proved algebraically.

For the third problem we have

Plot[{Sin[x]/(x (((Sin[x]))^2) + 1), (1/x), -(1/x)}, {x, 1, 
        30}, PlotStyle -> {Red, Blue, Blue}]

Try a different blue

Plot[{Sin[x]/(x (((Sin[x]))^2) + 1), (10/x), -(10/x)}, {x, 
        1, 30}, PlotStyle -> {Red, Blue, Blue}]

But, then a different domain and a different range gives a different picture

Plot[{Sin[x]/(x (((Sin[x]))^2) + 1), (10/x), -(10/x)}, {x, 
        1980, 2000}, PlotStyle -> {Red, Blue, Blue}, 
      PlotRange -> {-(10^(-2)), (10^(-2))}]

Doing plots indicates that NO blue function of the form: constant divided by x will work.  Can you discover which blue function will work? And then prove algebraically that it really works.