/usr/share/doc/axiom-doc/hypertex/callimits.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Limits</div>
  <hr/>
To compute a limit, you must specify a functional expression, a variable,
and a limiting value for that variable. If you do not specify a direction,
Axiom attempts to compute a two-sided limit.

Issue this to compute the limit of (x^2-2*x+2)/(x^2-1) as x approaches 1.
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="limit((x^2-3*x+2)/(x^2-1),x=1)" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
Sometimes the limit when approached from the left is different from the
limit from the right and, in this case, you may wish to ask for a one-sided
limit. Also, if you have a function that is only defined on one side of a
particular value, you can compute a one-sided limit.

The function log(x) is only defined to the right of zero, that is, for
x>0. Thus, when computing limits of functions involving log(x), you probably
want a "right-hand" limit.
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value='limit(x*log(x),x=0,"right")' />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
When you do not specify "right" or "left" as the optional fourth argument,
<a href="dboplimit.xhtml">limit</a> tries to compute a two-sided limit.
Here the limit from the left does not exist, as Axiom indicates when you
try to take a two-sided limit.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="limit(x*log(x),x=0)" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
A function can be defined on both sides of a particular value, but tend to
different limits as its variable approaches that value from the left and
from the right. We can construct an example of this as follows: Since 
sqrt(y^2) is simply the absolute value of y, the function sqrt(y^2)/y is
simply the sign (+1 or -1) of the nonzero real number y. Therefore,
sqrt(y^2)/y=-1 for y&#60;0 and sqrt(y^2)/y=+1 for y>0. This is what happens 
when we take the limit at y=0. The answer returned by Axiom gives both a
"left-handed" and a "right-handed" limit.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="makeRequest('p4');"
    value="limit(sqrt(y^2)/y,y=0)" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
Here is another example, this time using a more complicated function.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="makeRequest('p5');"
    value="limit(sqrt(1-cos(t))/t,t=0)" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
You can compute limits at infinity by passing either "plus infinity" or
"minus infinity" as the third argument of <a href="dboplimit.xhtml">limit</a>.
To do this, use the constants %plusInfinity and %minusInfinity.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="makeRequest('p6');"
    value="limit(sqrt(3*x^2+1)/(5*x),x=%plusInfinity)" />
  <div id="ansp6"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="makeRequest('p7');"
    value="limit(sqrt(3*x^2+1)/(5*x),x=%minusInfinity)" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
You can take limits of functions with parameters. As you can see, the limit
is expressed in terms of the parameters.
<ul>
 <li>
  <input type="submit" id="p8" class="subbut" 
    onclick="makeRequest('p8');"
    value="limit(sinh(a*x)/tan(b*x),x=0)" />
  <div id="ansp8"><div></div></div>
 </li>
</ul>
When you use <a href="dboplimit.xhtml">limit</a>, you are taking the limit
of a real function of a real variable. When you compute this, Axiom returns
0 because, as a function of a real variable, sin(1/z) is always between -1
and 1, so z*sin(1/z) tends to 0 as z tends to 0.
<ul>
 <li>
  <input type="submit" id="p9" class="subbut" 
    onclick="makeRequest('p9');"
    value="limit(z*sin(1/z),z=0)" />
  <div id="ansp9"><div></div></div>
 </li>
</ul>
However, as a function of a complex variable, sin(1/z) is badly behaved
near 0 (one says that sin(1/z) has an essential singularlity at z=0). When
viewed as a function of a complex variable, z*sin(1/z) does not approach any
limit as z tends to 0 in the complex plane. Axiom indicates this when we
call <a href="dbopcomplexlimit.xhtml">complexLimit</a>.
<ul>
 <li>
  <input type="submit" id="p10" class="subbut" 
    onclick="makeRequest('p10');"
    value="complexLimit(z*sin(1/z),z=0)" />
  <div id="ansp10"><div></div></div>
 </li>
</ul>
You can also take complex limits at infinity, that is, limits of a function
of z as z approaches infinity on the Riemann sphere. Use the symbol
%infinity to denote "complex infinity". As above, to compute complex limits
rather than real limits, use <a href="dbopcomplexlimit.xhtml">complexLimit</a>.
<ul>
 <li>
  <input type="submit" id="p11" class="subbut" 
    onclick="makeRequest('p11');"
    value="complexLimit((2+z)/(1-z),z=%infinity)" />
  <div id="ansp11"><div></div></div>
 </li>
</ul>
In many cases, a limit of a real function of a real variable exists when
the corresponding complex limit does not. This limit exists.
<ul>
 <li>
  <input type="submit" id="p12" class="subbut" 
    onclick="makeRequest('p12');"
    value="limit(sin(x)/x,x=%plusInfinity)" />
  <div id="ansp12"><div></div></div>
 </li>
</ul>
But this limit does not.
<ul>
 <li>
  <input type="submit" id="p13" class="subbut" 
    onclick="makeRequest('p13');"
    value="complexLimit(sin(x)/x,x=%infinity)" />
  <div id="ansp13"><div></div></div>
 </li>
</ul>
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axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / callimits.xhtml
