Limits

0. Introduction

In some sense, limits are the foundation of calculus.  In class we have discussed what a limit is and how to calculate some limits.  In this worksheet we will use Mathematica to explore the concepts of limits and to solve a bunch of problems.

1. Examples

First recall how to define a function in Mathematica: for instance, the function sin(x)/x is defined by,

f[x_]=Sin[x]/x

Note that you could use the Math Assistant palette to help input the function in a more standard notation.  Now let’s graph the function and take the limit as x approaches 0.

Plot[f[x],{x,-6,6}]

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

From the graph and the output of the Limit command we can see that the limit is 1.

How about the limit of ((|x|)/x)as x approaches 0?

Plot[((Abs[x])/x),{x,-3,3},PlotRange->{{-3,3},{-2,2}},PlotStyle->Directive[Blue,Thick]]

Limit[((Abs[x])/x),x->0]

Based on the graph, the limit should not be defined and Mathematica returns a value that indicates that the limit may not exist.  Mathematica can uses a one-sided limits. To specify the direction manually we would use the Direction option in the Limit command.  Note in the Document Center that the Limit command gives a limit from the plus side by

Limit[((Abs[x])/x),x->0,Direction->-1]

and a limit from the minus side by

Limit[((Abs[x])/x),x->0,Direction->1]

Thus, the two-sided limit DNE.

Name: 
PIN:

Tasks

By using the limit command described above (and possibly graphs), find the following limits in problems 1-4.  Write your answers on this worksheet and turn it in when finished.  Don’t forget to put your name and PIN at the top.

(a) (lim_(x->0))((sin(2x))/x)=		(b) (lim_(x->0))((sin(3x))/x)=		(c) (lim_(x->0))((sin(-5x))/x)=

(a) (lim_(x->(SuperPlus[0])))((sin(x))/(|x|))=		(b) (lim_(x->(SuperMinus[0])))((sin(x))/(|x|))=		(c) (lim_(x->0))((sin(x))/(|x|))=

(a) (lim_(x->0))((1-cos(x))/x)=		(b) (lim_(x->0))((1-cos(x))/(x^2))=		(c) (lim_(x->0))((1-cos(x))/(x^3))=

(a) (lim_(x->0))((1-cos(2x))/(x^2))=		(b) (lim_(x->0))((1-cos(7x))/(x^2))=		(c) (lim_(x->0))((1-cos(-4x))/(x^2))=

Define F(x)=(e^(3x))-1 and find

(a) (lim_(x->0))((F(x))/x)=		(b) (lim_(x->0))((F(2x))/x)=		(c) (lim_(x->0))((F(-7x))/x)=

Based on the above (first example, problem 1 and 5), if

(lim_(x->0))((F(x))/x)=L

then what would be the value of the following limit (where a is a constant):

(lim_(x->0))((F(a x))/x)=

Based on the above (problem 3b and 4), if

(lim_(x->0))((G(x))/(x^2))=M

then what would be the value of the following limit (where a is a constant):

(lim_(x->0))((G(a x))/(x^2))=