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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 "
" 0 "" {TEXT -1 46 "Lesson 2:  Continuity and Limits at Infinity I" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 257 9 "Example 1" }{TEXT -1 48 "\nFor th
e following function,  find the value of " }{TEXT 256 1 "a" }{TEXT -1 
37 " that makes the function continuous. " }}{PARA 0 "" 0 "" {TEXT -1 
79 "Plot the continuous function. Then take different values of the va
riable a and " }}{PARA 0 "" 0 "" {TEXT -1 45 "plot the associated disc
ontinuous functions. " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f(x) = a*x" "6
#/-%\"fG6#%\"xG*&%\"aG\"\"\"F'F*" }{TEXT -1 21 "  for x <= 2   and   \+
" }{XPPEDIT 18 0 "f(x) = a*x^2 + x + 1" "6#/-%\"fG6#%\"xG,(*&%\"aG\"\"
\"*$F'\"\"#F+F+F'F+F+F+" }{TEXT -1 12 "  for x > 2 " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 21 "Define the two pieces" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f1 \+
:= x->a*x; f2 := x->a*x^2+x+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f
1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&%\"aG\"\"\"9$F.F(F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#f2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*
&%\"aG\"\"\")9$\"\"#F/F/F1F/F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 89 "Substitute the break point x=2 into both functions, set t
hem equal, and then solve for a:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"a := solve(f1(2)=f2(2), a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG
#!\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f1(x); f2(x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"xG#!\"$\"\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"#!\"$F'F&F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 18 "Plot the function." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "f:= x -> piecewise( x <= 2, f1(x), f2(x));" }{TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operat
orG%&arrowGF(-%*piecewiseG6%19$\"\"#-%#f1G6#F0-%#f2GF4F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f(x), x = -5..5);" }
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f we'd gotten " }{TEXT 268 1 "a" }{TEXT -1 35 " wrong?  Suppose we tho
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11 "" 1 "" {XPPMATH 20 "6#>%\"aG!\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "f1(x); f2(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%
\"xG!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"!\"%F
&F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Plot the (discontinuou
s) function." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f:= x -> piecewise(
 x <= 2, f1(x), f2(x));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6%19$\"\"#
-%#f1G6#F0-%#f2GF4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 
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0 "" {TEXT -1 42 "Plot the graph of the following function. " }}{PARA 
0 "" 0 "" {TEXT -1 37 "Where is the function discontinuous? " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "q(t)  =  0  for
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}{PARA 0 "" 0 "" {TEXT -1 11 "        =  " }{XPPEDIT 18 0 "5*sqrt(t+3)
-8;" "6#,&*&\"\"&\"\"\"-%%sqrtG6#,&%\"tGF&\"\"$F&F&F&\"\")!\"\"" }
{TEXT -1 18 "   for 1 < t <= 6 " }}{PARA 0 "" 0 "" {TEXT -1 23 "      \+
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<= 1, 2*x^2, 1 < x and x <= 6, 5 * sqrt(x + 3) - 8, 7);" }{TEXT -1 0 "
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"xG6\"6$%)operatorG%&a
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{TEXT -1 38 "\nFor each of the following functions: " }}{PARA 0 "" 0 "
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{PARA 0 "" 0 "" {TEXT -1 70 "(ii) Use the plot to conjecture the limit
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G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "(iii) Determine prec
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