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+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Limits
+
+%  Author: Richard Jenks, Clifton J. Williamson
+%  Date created: 1 November 1989
+%  Date last updated: 25 May 1990
+
+\begin{page}{ExLimitBasic}{Computing Limits}
+\beginscroll
+To compute a limit, you must specify a functional expression,
+a variable, and a limiting value for that variable.
+For example, to compute the limit of (x**2 - 3*x + 2)/(x**2 - 1)
+as x approaches 1, issue the following command:
+\spadpaste{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+% answer := -1/2
+Since you have not specified a direction, \Language{} will attempt
+to compute a two-sided limit. Sometimes the limit when approached from
+the left is different from the limit from the right.
+\downlink{Example}{ExLimitTwoSided}. In this case, you may
+wish to ask for a one-sided limit.
+\downlink{How.  }{ExLimitOneSided}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitParameter}{Limits of Functions with Parameters}
+\beginscroll
+You may also take limits of functions with parameters. The limit
+will be expressed in terms of those parameters.
+Here's an example:
+\spadpaste{limit(sinh(a*x)/tan(b*x),x = 0)}
+% answer := a/b
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitOneSided}{One-sided Limits}
+\beginscroll
+If you have a function which is only defined on one side of a particular value,
+you may wish to compute a one-sided limit.
+For instance, the function \spad{log(x)} is only defined to the right of zero,
+i.e. for \spad{x > 0}.
+Thus, when computing limits of functions involving \spad{log(x)}, you probably
+will want a 'right-hand' limit.
+Here's an example:
+\spadpaste{limit(x * log(x),x = 0,"right")}
+% answer := 0
+When you do not specify \spad{right} or \spad{left} as an optional fourth
+argument, the function \spadfun{limit} will try to compute a two-sided limit.
+In the above case, the limit from the left does not exist, as \Language{}
+will indicate when you try to take a two-sided limit:
+\spadpaste{limit(x * log(x),x = 0)}
+% answer := [left = "failed",right = 0]
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitTwoSided}{Two-sided Limits}
+\beginscroll
+A function may be defined on both sides of a particular value, but will
+tend to different limits as its variable tends to that value from the
+left and from the right.
+We can construct an example of this as follows:
+Since { \em sqrt(y**2)} is simply the absolute value of \spad{y},
+the function \spad{sqrt(y**2)/y}
+is simply the sign (+1 or -1) of the real number \spad{y}.
+Therefore, \spad{sqrt(y**2)/y = -1} for \spad{y < 0} and
+\spad{sqrt(y**2)/y = +1} for \spad{y > 0}.
+Watch what happens when we take the limit at \spad{y = 0}.
+\spadpaste{limit(sqrt(y**2)/y,y = 0)}
+% answer := [left = -1,right = 1]
+The answer returned by \Language{} gives both a 'left-hand' and a 'right-hand'
+limit.
+Here's another example, this time using a more complicated function:
+\spadpaste{limit(sqrt(1 - cos(t))/t,t = 0)}
+% answer := [left = -sqrt(1/2),right = sqrt(1/2)]
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitInfinite}{Limits at Infinity}
+\beginscroll
+You can compute limits at infinity by passing either 'plus infinity'
+or 'minus infinity' as the third argument of the function \spadfun{limit}.
+To do this, use the constants \spad{\%plusInfinity} and \spad{\%minusInfinity}.
+Here are two examples:
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitRealComplex}{Real Limits vs. Complex Limits}
+\beginscroll
+When you use the function \spadfun{limit}, you will be taking the limit of a real
+function of a real variable.
+For example, you can compute
+\spadpaste{limit(z * sin(1/z),z = 0)}
+\Language{} returns \spad{0} because as a function of a real variable
+\spad{sin(1/z)} is always between \spad{-1} and \spad{1}, so \spad{z * sin(1/z)}
+tends to \spad{0} as \spad{z} tends to \spad{0}.
+However, as a function of a complex variable, \spad{sin(1/z)} is badly
+behaved around \spad{0}
+(one says that \spad{sin(1/z)} has an 'essential singularity' at \spad{z = 0}).
+When viewed as a function of a complex variable, \spad{z * sin(1/z)}
+does not approach any limit as \spad{z} tends to \spad{0} in the complex plane.
+\Language{} indicates this when we call the function \spadfun{complexLimit}:
+\spadpaste{complexLimit(z * sin(1/z),z = 0)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitComplexInfinite}{Complex Limits at Infinity}
+\beginscroll
+You may also take complex limits at infinity, i.e. limits of a function of
+\spad{z} as \spad{z} approaches infinity on the Riemann sphere.
+Use the symbol \spad{\%infinity} to denote `complex infinity'.
+Also, to compute complex limits rather than real limits, use the
+function \spadfun{complexLimit}.
+Here is an example:
+\spadpaste{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+In many cases, a limit of a real function of a real variable will exist
+when the corresponding complex limit does not.
+For example:
+\spadpaste{limit(sin(x)/x,x = \%plusInfinity)}
+\spadpaste{complexLimit(sin(x)/x,x = \%infinity)}
+\endscroll
+\autobuttons\end{page}
+