March 24, 2006 7:10 am
Larson ch 1.4, p. 75 - 78.
14.

In[1]:=

y = (Sqrt[x] - 2)/(x - 4)

Out[1]=

(-2 + x^(1/2))/(-4 + x)

In[2]:=

Limit[y, x4, Direction -1]

Out[2]=

1/4

In[4]:=

p1 = Plot[y, {x, 0, 5}, PlotStyleBlue] ;

[Graphics:HTMLFiles/index_6.gif]

In[5]:=

Clear[y, p1]


15.

In[6]:=

y = Abs[x]/x

Out[6]=

Abs[x]/x

In[7]:=

Limit[y, x0]

Out[7]=

1

In[8]:=

Plot[y, {x, -5, 5}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_13.gif]

Interesting. The limit fails to exist since the right and left limits do not agree. Yet Mathematica gives an answer of 1. This at least illustrates the value of obtaining a plot as well as an analytic solution.
16.

In[9]:=

Clear[y, p1]

In[10]:=

y = Abs[x - 2]/(x - 2)

Out[10]=

Abs[-2 + x]/(-2 + x)

In[11]:=

Limit[y, x2]

Out[11]=

1

In[14]:=

Limit[y, x2, Direction0]

Out[14]=

Indeterminate

Super! Setting the Direction to zero is equivalent to computing both the right and left sided limits. Let me try a quick check.

In[15]:=

Clear[y]

In[16]:=

y = x

Out[16]=

x

In[17]:=

Limit[y, x0, Direction0]

Out[17]=

0

Perfect.
17.

In[18]:=

Clear[y]

In[19]:=

y = ((1/(x + dx)) - 1/x)/dx

Out[19]=

(-1/x + 1/(dx + x))/dx

In[20]:=

Limit[y, dx0, Direction1]

Out[20]=

-1/x^2


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