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{SCP_R 1 0 83{COM_R 2 0{TEXT 1 0 ""}}{COM_R 3 0{TEXT 14 28 "Limit
s and Series with Maple"}}{COM_R 4 0{TEXT 1 0 ""}}{COM_R 5 0{TEXT
 1 0 ""}}{COM_R 6 0{TEXT 1 305 "Maple has a deep knowledge of man
y operations from mathematical analysis.   It can integrate, diff
erentiate, find limits, perform several types of integral transfo
rmations, and expand functions in a variety of different series. \+
 This worksheet is a demonstration of Maple's knowledge of limits
 and series."}}{COM_R 7 0{TEXT 1 0 ""}}{COM_R 8 0{TEXT 1 200 "In \+
general, Maple calculates limits by examining the function's seri
es expansion.  Aside from continuous functions, Maple can calcula
te limits at singularities and one sided limits at discontinuitie
s."}}{COM_R 9 0{TEXT 1 0 ""}}{INP_R 10 0 "> "{TEXT 0 16 "eq1 := s
in(x)/x;"}}{OUT_R 11 0 10{DAG :3n3\`eq1`*5(3n3\`sin`,2n3\`x`j2x00
01p8i2x0001}}{INP_R 12 0 "> "{TEXT 0 15 "limit(eq1,x=0);"}}{OUT_R
 13 0 12{DAG j2x0001}}{COM_R 14 0{TEXT 1 0 ""}}{COM_R 15 0{TEXT 1
 60 "In this case, the function had a removable singularity at 0.
"}}{COM_R 16 0{TEXT 1 0 ""}}{COM_R 17 0{TEXT 1 190 "This next exa
mple is a linear  fractional transformation.  Functions of this t
ype have , in general, only one singularity on the complex plane.
  This one is bounded at the point at infinity."}}{COM_R 18 0
{TEXT 1 0 ""}}{INP_R 19 0 "> "{TEXT 0 23 "eq2 := (2*x+3)/(7*x+5);
"}}{OUT_R 20 0 19{DAG :3n3\`eq2`*5+5n3\`x`j2x0002j2x0003j2x0001pB
+5p5j2x0007j2x0005pBi2x0001}}{INP_R 21 0 "> "{TEXT 0 22 "limit(eq
2,x=infinity);"}}{OUT_R 22 0 21{DAG /3j2x0002j2x0007}}{COM_R 23 0
{TEXT 1 0 ""}}{COM_R 24 0{TEXT 1 52 "This function also has a rem
ovable singularity at 1."}}{COM_R 25 0{TEXT 1 0 ""}}{INP_R 26 0 "
> "{TEXT 0 41 "eq3 := (x^2-2*x+1)/(x^4+3*x^3-7*x^2+x+2);"}}{OUT_R
 27 0 26{DAG :3n3\`eq3`*5+7*3n3\`x`j2x0002j2x0001p6i2x0002pApApA+
B*3p6j2x0004pA*3p6j2x0003p1Ap5i2x0007p6pAp8pAi2x0001}}{INP_R 28 0
 "> "{TEXT 0 14 "subs(x=1,eq3);"}}{OUT_R 29 0 28{TEXT 2 24 "Error
, division by zero\012"}}{INP_R 30 0 "> "{TEXT 0 15 "limit(eq3,x=
1);"}}{OUT_R 31 0 30{DAG /3j2x0001j2x0008}}{COM_R 32 0{TEXT 1 0 "
"}}{COM_R 33 0{TEXT 1 266 "This next function is quite difficult \+
to analyze near r=0.  The limit of each term of the difference is
 infinite and if the terms are combined into a single fraction, t
he denominator has 0 derivatives of all orders at 0, making l'Hop
ital's rule useless in this case."}}{COM_R 34 0{TEXT 1 0 ""}}
{INP_R 35 0 "> "{TEXT 0 113 "Limit((2/(a^2-b^2)^3/r^6-1/(a^2-b^2)
^2/r^4)/exp(b^2*r^2)-(2/(a^2-b^2)^3/r^6+1/(a^2-b^2)^2/r^4)/exp(a^
2*r^2),r=0);"}}{OUT_R 36 0 35{DAG (3n4\`Limit`,3+5*5+5*5+5*3n3\`a
`j2x0002j2x0001*3n3\`b`pDi2x0001i2x0003n3\`r`i2x0006pD*5p9i2x0002
p19i2x0004p15pF(3n3\`exp`,2*5p12pDp19pDp15pF*5+5p8pDp1EpFpF(3p28,
2*5pBpDp19pDp15p15=3p19j2x0000}}{INP_R 37 0 "> "{TEXT 0 9 "value(
\");"}}{OUT_R 38 0 37{DAG /3i2x0001j2x0006}}{COM_R 39 0{TEXT 1 0 
""}}{COM_R 40 0{TEXT 1 157 "Maple's limit is over the real line b
y default.  If we want the complex limit, we must specify the opt
ion 'complex'.  This explains the following  behaviour:"}}{COM_R 
41 0{TEXT 1 0 ""}}{INP_R 42 0 "> "{TEXT 0 15 "limit(1/x,x=0);"}}
{OUT_R 43 0 42{DAG n5\`undefined`}}{COM_R 44 0{TEXT 1 0 ""}}
{COM_R 45 0{TEXT 1 111 "Maple's answer is undefined since the lim
it from the right is positive and the limit from the left is nega
tive."}}{COM_R 46 0{TEXT 1 0 ""}}{INP_R 47 0 "> "{TEXT 0 23 "limi
t(1/x,x=0,complex);"}}{OUT_R 48 0 47{DAG n5\`infinity`}}{COM_R 49
 0{TEXT 1 0 ""}}{COM_R 50 0{TEXT 1 126 "Maple's answer is infinit
y here since there is only one  point at infinity in complex anal
ysis and 1/x has a simple pole at 0."}}{COM_R 51 0{TEXT 1 0 ""}}
{COM_R 52 0{TEXT 1 44 "We can also ask for one sided, real  limit
s."}}{COM_R 53 0{TEXT 1 0 ""}}{INP_R 54 0 "> "{TEXT 0 21 "limit(1
/x,x=0,right);"}}{OUT_R 55 0 54{DAG n5\`infinity`}}{INP_R 56 0 ">
 "{TEXT 0 20 "limit(1/x,x=0,left);"}}{OUT_R 57 0 56{DAG +3n5\`inf
inity`i2x0001}}{COM_R 58 0{TEXT 1 0 ""}}{COM_R 59 0{TEXT 1 214 "A
s mentioned before, Maple uses series to find many of its limits.
  Maple can calculate Taylor series, Laurent series, Chebyshev se
ries, and Puiseux series.  Here is an example of the easiest type
, a Taylor series."}}{COM_R 60 0{TEXT 1 0 ""}}{INP_R 61 0 "> "
{TEXT 0 14 "eq5:=x*sin(x);"}}{OUT_R 62 0 61{DAG :3n3\`eq5`*5n3\`x
`j2x0001(3n3\`sin`,2p4p6}}{COM_R 63 0{TEXT 1 0 ""}}{COM_R 64 0
{TEXT 1 142 "The function series tries to find the best type of s
eries for the job.  If the function appears to be analytic, it ca
lculates a Taylor series."}}{COM_R 65 0{TEXT 1 0 ""}}{INP_R 66 0 
"> "{TEXT 0 16 "series(eq5,x=0);"}}{OUT_R 67 0 66{DAG _8n3\`x`j2x
0001x2/3i2x0001j2x0006x4(3n3\`O`,2p3x6}}{COM_R 68 0{TEXT 1 0 ""}}
{COM_R 69 0{TEXT 1 135 "We can specify the order of the series wi
th the global variable Order (default is 6) or by specifying the \+
desired order as an argument."}}{COM_R 70 0{TEXT 1 0 ""}}{INP_R 
71 0 "> "{TEXT 0 19 "series(eq5,x=0,10);"}}{OUT_R 72 0 71{DAG _Cn
3\`x`j2x0001x2/3i2x0001j2x0006x4/3p3j2x0120x6/3p7j2x5040x8(3n3\`O
`,2p3xA}}{COM_R 73 0{TEXT 1 0 ""}}{COM_R 74 0{TEXT 1 81 "If the s
eries has a pole at the point of expansion, Maple finds a Laurent
 series."}}{COM_R 75 0{TEXT 1 0 ""}}{INP_R 76 0 "> "{TEXT 0 18 "e
q6:=1/x^5*sin(x);"}}{OUT_R 77 0 76{DAG :3n3\`eq6`*5n3\`x`i2x0005(
3n3\`sin`,2p4j2x0001}}{INP_R 78 0 "> "{TEXT 0 16 "series(eq6,x=0)
;"}}{OUT_R 79 0 78{DAG _An3\`x`j2x0001xFFFFFFFC/3i2x0001j2x0006xF
FFFFFFE/3p3j2x0120x0(3n3\`O`,2p3x2}}{INP_R 80 0 "> "{TEXT 0 61 "e
q7 := (x^2 + 2*x - 3) / (x^4 - 11*x^3 + 42*x^2 - 68*x + 40);"}}
{OUT_R 81 0 80{DAG :3n3\`eq7`*5+7*3n3\`x`j2x0002j2x0001p6p8i2x000
3pApA+B*3p6j2x0004pA*3p6j2x0003i2x0011p5j2x0042p6i2x0068j2x0040pA
i2x0001}}{INP_R 82 0 "> "{TEXT 0 16 "series(eq7,x=2);"}}{OUT_R 83
 0 82{DAG _10+5n3\`x`j2x0001i2x0002p4/3i2x0005j2x0003xFFFFFFFD/3i
2x0023j2x0009xFFFFFFFE/3i2x0032j2x0027xFFFFFFFF/3p16j2x0081x0/3p1
6j2x0243x1/3p16j2x0729x2(3n3\`O`,2p4x3}}{INP_R 84 0 "> "{TEXT 0 1
 "\012"}}}{END}
