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 0 52{COM_R 2 0{TEXT 7 25 "An Introduction to Limits"}}{COM_R 3 0
{TEXT 1 31 "\015Copyright 1993 by J. S. Devitt"}}{COM_R 4 0{TEXT 
1 0 ""}}{INP_R 5 0 "> "{TEXT 0 14 "with(student):"}}{COM_R 6 0
{TEXT 1 0 ""}}{COM_R 7 0{TEXT 1 63 "Limits become important when \+
straight forward evaluation fails."}}{COM_R 8 0{TEXT 1 8 "Example
:"}}{COM_R 9 0{TEXT 1 0 ""}}{INP_R 10 0 "> "{TEXT 0 26 "f := (x) \+
-> (x^2-1)/(x+1);"}}{OUT_R 11 0 10{DAG :3n3\`f`@8,2n3\`x`,1,3n5\`
operator`n4\`arrow`p7*5+5*3a2x0001j2x0002j2x0001i2x0001p18p18+5p1
4p18p18p18p1Ap7p7}}{INP_R 12 0 "> "{TEXT 0 5 "f(3);"}}{OUT_R 13 0
 12{DAG j2x0002}}{INP_R 14 0 "> "{TEXT 0 5 "f(x);"}}{OUT_R 15 0 
14{DAG *5+5*3n3\`x`j2x0002j2x0001i2x0001p7p7+5p3p7p7p7p9}}{INP_R 
16 0 "> "{TEXT 0 6 "f(-1);"}}{OUT_R 17 0 16{TEXT 2 31 "Error, (in
 f) division by zero\012"}}{COM_R 18 0{TEXT 1 58 "Except for this
 one point (t=-1), f is defined everywhere."}}{COM_R 19 0{TEXT 1 
35 "What should the value of f(-1) be ?"}}{COM_R 20 0{TEXT 1 51 "
To answer this question, we look at the graph of f."}}{INP_R 21 0
 "> "{TEXT 0 16 "plot(f,-10..10);"}}{COM_R 22 0{TEXT 1 0 ""}}
{COM_R 23 0{TEXT 1 80 "The graph suggests that f is essentially t
he same as the function g  defined as:"}}{COM_R 24 0{TEXT 1 15 "g
: (x) -> (x-1)"}}{COM_R 25 0{TEXT 1 56 "This is easy to see if we
 factor the numerator of  f(x);"}}{COM_R 26 0{TEXT 1 0 ""}}{INP_R
 27 0 "> "{TEXT 0 5 "f(x);"}}{OUT_R 28 0 27{DAG *5+5*3n3\`x`j2x00
02j2x0001i2x0001p7p7+5p3p7p7p7p9}}{INP_R 29 0 "> "{TEXT 0 17 "fac
tor(numer(\"));"}}{OUT_R 30 0 29{DAG *5+5n3\`x`j2x0001i2x0001p4p4
+5p2p4p4p4p4}}{COM_R 31 0{TEXT 1 129 "The common factor   (x + 1)
 can be cancelled and,  except for x = -1, the new expression wil
l have the same value of the old one."}}{COM_R 32 0{TEXT 1 97 "Th
e value we expect  at   x = -1  is determined by examining the fu
nction value at nearby points."}}{COM_R 33 0{TEXT 1 13 "We say th
at: "}}{COM_R 34 0{TEXT 1 95 "         the limit of f(x), as x ap
proaches -1,   is    -2.\015We write this in the following way\01
5"}}{INP_R 35 0 "> "{TEXT 0 38 "Limit('f'(x),x=-1) = limit(f(x),x
=-1);"}}{OUT_R 36 0 35{DAG =3(3n4\`Limit`,3(3n3\`f`,2n3\`x`=3pAi2
x0001i2x0002}}{COM_R 37 0{TEXT 1 0 ""}}{COM_R 38 0{TEXT 1 44 "Usi
ng Limits to Approximate a rate of change"}}{COM_R 39 0{TEXT 1 99
 "Given two points on a curve:  [a,f(a)] and [b,f(b)]\012the slop
e of the straight line between them is:"}}{INP_R 40 0 "> "{TEXT 0
 18 "(f(b)-f(a))/(b-a);"}}{OUT_R 41 0 40{DAG *5+5*5+5*3n3\`b`j2x0
002j2x0001i2x0001p9p9+5p5p9p9p9pBp9*5+5*3n3\`a`p7p9pBp9p9+5p19p9p
9p9pBpBp9+5p5p9p19pBpB}}{COM_R 42 0{TEXT 1 53 "If one of the end \+
points (b) is x this is written as:"}}{INP_R 43 0 "> "{TEXT 0 12 
"subs(b=x,\");"}}{OUT_R 44 0 43{DAG *5+5*5+5*3n3\`x`j2x0002j2x000
1i2x0001p9p9+5p5p9p9p9pBp9*5+5*3n3\`a`p7p9pBp9p9+5p19p9p9p9pBpBp9
+5p5p9p19pBpB}}{COM_R 45 0{TEXT 1 56 "As   x   gets closer to   a
  , we are computing a limit."}}{INP_R 46 0 "> "{TEXT 0 13 "Limit
(\",x=a);"}}{OUT_R 47 0 46{DAG (3n4\`Limit`,3*5+5*5+5*3n3\`x`j2x0
002j2x0001i2x0001pEpE+5pApEpEpEp10pE*5+5*3n3\`a`pCpEp10pEpE+5p1Ep
EpEpEp10p10pE+5pApEp1Ep10p10=3pAp1E}}{COM_R 48 0{TEXT 1 43 "The v
alue of this expression is denoted as:"}}{COM_R 49 0{TEXT 1 5 "f'
(a)"}}{COM_R 50 0{TEXT 1 58 "This is \"The derivative of f\", eva
luated at the point \"a\"."}}{INP_R 51 0 "> "{TEXT 0 9 "value(\")
;"}}{OUT_R 52 0 51{DAG j2x0001}}{INP_R 53 0 "> "{TEXT 0 0 ""}}}
{END}
