Worksheet 2, Math 151

CALCULATING LIMITS GRAPHICALLY

Introduction

Goals: In this worksheet we will analyze the limits of functions with animated graphs and Mathematica's Limit command.

Examples

Consider the following limits:
1. ({{lim}, {x->0}})((sin(x))/x)         2. ({{lim}, {x->0}})sin((1/x))
To study the limit in 1, we first define the function f(x)=((sin(x))/x) in Mathematica.

f[x_]:=Sin[x]/x

An animation will now be used to estimate the limit geometrically. After entering the following sequence of statements, a little GUI will appear that allows you to interactively visualize the limiting behavior of this function. You can either use your mouse to move the slider, or you can click on the plus symbol at the end of the slider line to open up more controls.

Manipulate[Plot[f[x],{x,-10,-10+a},PlotRange ->{{-11,11}, {-0.3,1.1}}],{a,0.001,10}]

Again, if you click and drag the slider next to the parameter a at the top of the GUI, you will get a reasonable estimate for the limit at x = 0.

Manipulate[Plot[f[x],{x,10-a,10},PlotRange ->{{-11,11}, {-0.3,1.1}}],{a,0.001,10}]

To compute the limit exactly, we use Mathematica's "Limit" command.

Limit[f[x],x -> 0]

For the limit in 2, we define the function g(x)=sin(1/x) in Mathematica, and then plot its graph over the domain [-π,π]. Note that when x is near zero, the graph oscillates wildly and does not get close to any single value.

g[x_]:=Sin[1/x]

Plot[g[x],{x,-Pi,Pi}]

The oscillatory nature of this function becomes evern more apparent when attention is focused on that portion of its graph very close to the origin.

Plot[g[x],{x,-0.1,0.1}]

We conclude from this graphical analysis that the limit does not exist. However, we will still attempt to compute this limit using Mathematica.

Limit[g[x],x-> 0]

This is Mathematica's way of telling us that the limit could potentially take every value in the real interval from -1 to +1.
Since a limit must be unique if it exists, it follows that ({{lim}, {x->0}})sin((1/x)) does not exist.

Exercises

Exercise 1

(a) Enter a sequence of commands to animate the limit ({{lim}, {x->0}})((1-cos(x))/((πx)^2)).

(b) Find the exact value of ({{lim}, {x->0}})((1-cos(x))/((πx)^2)) by using Mathematica's Limit command.

Exercise 2

Let f(x)=((3sin(x)-2xcos((x^3)))/(|x((|)^c))). For each choice of c=(1/2),(3/4),1,(5/4),(3/2), do the following:

(a) Define the function f in Mathematica.

(b) Plot the graph of the function y=f(x) near the origin. Be certain to choose a suitable range for y in each case so that the characteristic shape of the graph is evident.

(c) From your plot, classify the function f as even or odd and then guess the value of the limit of the function at x=0.

(d) Evaluate the limit symbolically using Mathematica.

Exercise 3

Let f(x)=((3sin(x)-2xcos((x^3)))/(|x((|)^c))). From your analysis in problem 2, what do you suppose happens to ({{lim}, {x->0}})f(x) whenever c is a rational number <+1? Whenever c is a rational number >+1?
Give a motivation for your conjecture by speculating on the relative rates at which the numerator and denominator in this limit problem tend to zero as c varies.

Exercise 4

In this problem you will compare the behavior of the functions f(x)=x sin((1/(x^3))) and g(x)=(x^2)sin((1/(x^3))) near the origin.

(a) Define the functions f and g in Mathematica.

f[x_]:=x Sin[1/x^3]

(b) Execute the following plot statement and then comment on that you see

Plot[{f[x],Abs[x],-Abs[x]},{x,-1/20,1/20},PlotStyle->{{Magenta},{Blue,Thick},{Orange,Thick}}]

(c) Create a plot similar to that from part (b) which displays the graph of y=g(x) and its two bounding curves.

(d) Use Mathematica to compute the limits: ({{lim}, {x->0}})f(x) and  ({{lim}, {x->0}})g(x).

(e) Verify the limits obtained from part (d) using the analytical properties of limits.