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RATIONS: Unit 28" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{PARA 259 "" 0 "" {TEXT 336 23 "Dr. Wlodzislaw Kostecki" }}{PARA 260 "" 0 " " {TEXT -1 53 "The Papua New Guinea University of Technology (PNGUT)" }}{PARA 260 "" 0 "" {TEXT -1 54 "Department of Electrical and Communic ation Engineering" }}{PARA 260 "" 0 "" {TEXT -1 20 "Lae, Morobe Provin ce" }}{PARA 260 "" 0 "" {TEXT -1 16 "Papua New Guinea" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 335 41 "Copyright \251 200 0 by Wlodzislaw Kostecki" }}{PARA 259 "" 0 "" {TEXT 337 19 "All right s reserved" }}{PARA 2 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 338 67 "-------------------------------------------------------------- -----" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 4 "(28)" }{TEXT 258 1 " " }{TEXT 259 39 "Limits of matrice s comprising functions" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 386 10 "OBJECTIVES" }{TEXT 387 1 ":" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 388 1 "\225" }{TEXT -1 34 " To provide a definition of the " }{TEXT 392 5 "limit" }{TEXT -1 51 " of a matrix in the form of a symbolic ex pression." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 391 1 "\225" }{TEXT -1 20 " To point out that " }{TEXT 390 5 "Maple" }{TEXT -1 39 " can compute \+ limits of matrices where " }{TEXT 393 13 "indeterminate" }{TEXT -1 21 " forms are involved." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 394 1 "\225 " }{TEXT -1 95 " To exemplify computations of limits of matrices for \+ all the types of the indeterminate forms." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 395 1 "\225" }{TEXT -1 98 " To provide five examples of limits \+ of matrices containing functions as their variable tends to " }{TEXT 396 8 "infinity" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 397 1 "\225" }{TEXT -1 98 " To provide five examples of limits of matrice s containing functions as their variable tends to " }{XPPMATH 20 "6#% %zeroG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "If some o r all elements of a matrix [" }{TEXT 385 1 "A" }{TEXT -1 7 "] are " } {TEXT 378 9 "functions" }{TEXT -1 5 " of " }{TEXT 379 3 "one" }{TEXT -1 11 " variable " }{TEXT 380 1 "t" }{TEXT 381 1 "," }{TEXT -1 44 " \+ it is convenient to denote the matrix as [" }{TEXT 382 1 "A" }{TEXT 384 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 383 1 ")" }{TEXT -1 46 " ] when considering the limit of such a matrix." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The limit of a matrix [" }{TEXT 269 1 "A" }{TEXT 306 1 "(" }{XPPEDIT 18 0 "t" "6#% \"tG" }{TEXT 305 1 ")" }{TEXT -1 37 "] whose elements are functions of a " }{TEXT 321 4 "real" }{TEXT -1 12 " variable " }{XPPEDIT 18 0 " t" "6#%\"tG" }{TEXT -1 7 ", as " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT -1 14 " approaches " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 64 ", is defined to be the matrix of the same order with elements " }{XPPEDIT 18 0 "Limit(a[ij](t),t=a)" "6#-%&LimitG6$-&%\"aG6#%#ijG6#%\" tG/F,F(" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "For example, consider a " }{TEXT 322 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 267 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 323 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 268 1 "A" }{TEXT 308 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 307 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A(t) := matrix(2, 3, [a[11](t), a[12](t), a[13](t), \+ a[21](t), a[22](t), a[23](t)]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%-&%\"aG6#\"#6F&-&F/6#\" #7F&-&F/6#\"#8F&7%-&F/6#\"#@F&-&F/6#\"#AF&-&F/6#\"#BF&" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 271 5 "limit" }{TEXT -1 35 " function applied to each eleme nt " }{XPPEDIT 18 0 "a[ij](t)" "6#-&%\"aG6#%#ijG6#%\"tG" }{TEXT -1 17 " of the matrix [" }{TEXT 270 1 "A" }{TEXT 310 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 309 1 ")" }{TEXT -1 45 "] attempts to compute t he limiting value of " }{XPPEDIT 18 0 "a[ij](t)" "6#-&%\"aG6#%#ijG6#% \"tG" }{TEXT -1 6 " as " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 48 " approaches a given value of the variable, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "`limit(A)` := map(Limit, A(t), t=a) : Limit ('A(t)', t=a) = matrix(`limit(A)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%&LimitG6$-%\"AG6#%\"tG/F*%\"aG-%'matrixG6#7$7%-F%6$-&F,6#\"#6F)F+ -F%6$-&F,6#\"#7F)F+-F%6$-&F,6#\"#8F)F+7%-F%6$-&F,6#\"#@F)F+-F%6$-&F,6# \"#AF)F+-F%6$-&F,6#\"#BF)F+" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This operation can be displayed \+ in \"like-in-a-book\" form, viz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Limit(A(t), t=a) = matrix(`limit(A)`) ;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%'matrixG6#7$7%-&%\"aG6#\"#66#%\"tG -&F.6#\"#7F1-&F.6#\"#8F17%-&F.6#\"#@F1-&F.6#\"#AF1-&F.6#\"#BF1/F2F.-F( 6#7$7%-F%6$F,FH-F%6$F3FH-F%6$F7FH7%-F%6$F " 0 "" {MPLTEXT 1 0 24 "Limit(f(t), t=a, left) \+ ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6%-%\"fG6#%\"tG/F)%\"aG% %leftG" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "If the parameter is " }{TEXT 280 4 "left" }{TEXT -1 4 " or " }{TEXT 281 5 "right" }{TEXT -1 18 ", the limit is a " } {TEXT 282 11 "directional" }{TEXT -1 58 " limit, taken from the left \+ or right of the limit point " }{TEXT 285 1 "a" }{TEXT -1 15 ", respec tively." }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "If no additional parameter is specified, " }{TEXT 402 5 "Maple" }{TEXT -1 22 " considers the limit " }{TEXT 398 4 "real " }{TEXT -1 7 " and " }{TEXT 403 13 "bidirectional" }{TEXT -1 25 ". \+ Where the limit point " }{TEXT 401 1 "a" }{TEXT -1 4 " is " }{TEXT 399 8 "infinity" }{TEXT -1 4 " or " }{TEXT 400 9 "-infinity" }{TEXT -1 55 ", the limit of the function is taken from the left to " } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 24 " or from the r ight to " }{XPPEDIT 18 0 "-infinity" "6#,$%)infinityG!\"\"" }{TEXT 404 1 "," }{TEXT -1 15 " respectively." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Apart from the ma trices whose limits are obtainable by computing " }{XPPEDIT 18 0 "f(a )" "6#-%\"fG6#%\"aG" }{TEXT -1 20 " for each element " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT 389 1 "," }{TEXT -1 2 " " }{TEXT 272 5 "Maple" }{TEXT -1 40 " can compute limits where substitution " }{XPPEDIT 18 0 "t=a" "6#/%\"tG%\"aG" }{TEXT -1 42 " results in indete rminate forms of type " }{XPPEDIT 18 0 "0/0" "6#*&\"\"!\"\"\"F$!\"\" " }{TEXT -1 1 " " }{TEXT 261 1 "," }{TEXT -1 2 " " }{XPPEDIT 18 0 "in finity/infinity" "6#*&%)infinityG\"\"\"F$!\"\"" }{TEXT -1 1 " " } {TEXT 262 1 "," }{TEXT -1 2 " " }{XPPEDIT 18 0 "0*infinity" "6#*&\"\" !\"\"\"%)infinityGF%" }{TEXT 260 1 "," }{TEXT -1 2 " " }{XPPEDIT 18 0 "infinity-infinity" "6#,&%)infinityG\"\"\"F$!\"\"" }{TEXT 263 1 "," }{TEXT -1 2 " " }{XPPEDIT 18 0 "0^0" "6#*$\"\"!F$" }{TEXT 264 1 "," } {TEXT -1 2 " " }{XPPEDIT 18 0 "infinity^0" "6#*$%)infinityG\"\"!" } {TEXT 265 1 "," }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1^infinity" "6#) \"\"\"%)infinityG" }{TEXT 266 1 "." }{TEXT -1 28 " In computing such \+ limits, " }{TEXT 273 5 "Maple" }{TEXT -1 5 " uses" }{TEXT 287 5 " L \222H" }{TEXT -1 1 "\364" }{TEXT 274 6 "pital\222" }{TEXT -1 1 "s" } {TEXT 286 2 " " }{TEXT -1 74 "rule. The following examples include al l types of the indeterminate forms." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 80 "Examples of limits of matrices containing functions as their variable tends to " }{TEXT 283 8 "infinity" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 9 "Example 1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " } {XPPEDIT 18 0 "Limit(A(t),t=infinity)" "6#-%&LimitG6$-%\"AG6#%\"tG/F)% )infinityG" }{TEXT -1 9 " for a " }{TEXT 324 1 "(" }{XPPEDIT 18 0 "2 " "6#\"\"#" }{TEXT 289 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 325 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 290 1 "A" }{TEXT 312 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 311 1 ")" }{TEXT -1 10 "] given as " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "a[11](t) := (1+1/t)^t \+ : a[12](t) := (1+2/t)^t : a[21](t) := (t/(1+t))^t : a[22](t) := \+ sin((t/(t-1))^t) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "A(t) \+ := matrix([ [a[11](t), a[12](t)], [a[21](t), a[22](t)]]) : 'A(t)' = \+ A(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$ 7$),&\"\"\"F/*&F/F/F'!\"\"F/F'),&F/F/*&\"\"#F/F'F1F/F'7$)*&F'F/,&F/F/F 'F/F1F'-%$sinG6#)*&F'F/,&F'F/F/F1F1F'" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`lim A(t)` \+ := map(limit, A(t), t=infinity) : Limit('A(t)', t=infinity) = matrix (`lim A(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"AG6# %\"tG/F*%)infinityG-%'matrixG6#7$7$-%$expG6#\"\"\"-F36#\"\"#7$-F36#!\" \"-%$sinG6#F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-point evaluation yields" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`lim A(t)` := evalf(matrix(` lim A(t)`)) : Limit('A(t)', t=infinity) = matrix(`lim A(t)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"AG6#%\"tG/F*%)infinity G-%'matrixG6#7$7$$\"+G=G=F!\"*$\"+*4c!*Q(F47$$\"+7WzyO!#5$\"+4H\"y5%F: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 9 "Example 2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "Limit(B( t),t=infinity)" "6#-%&LimitG6$-%\"BG6#%\"tG/F)%)infinityG" }{TEXT -1 9 " for a " }{TEXT 326 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 292 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 327 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 293 1 "B" }{TEXT 314 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 313 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "b[11](t) := (t+(-1)^t)/(t-(-1)^t) \+ : b[12](t) := (3^(t+2)+5^(t+2))/(3^t-5^t) : b[13](t) := (2*t-1)/(3 *t+1) : b[21](t) := t*(sqrt(t^2+3) - t) : b[22](t) := t*sin(1/t) \+ : b[23](t) := (1-t)/(1+t) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "B(t) := matrix([ [b[11](t), b[12](t), b[13](t)], [b[21](t), b[2 2](t), b[23](t)]]) : 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG-%'matrixG6#7$7%*&,&F'\"\"\")!\"\"F'F/F/,&F'F/F0F1 F1*&,&)\"\"$,&F'F/\"\"#F/F/)\"\"&F7F/F/,&)F6F'F/)F:F'F1F1*&,&F'F8F/F1F /,&F'F6F/F/F17%*&F'F/,&*$-%%sqrtG6#,&*$)F'F8F/F/F6F/F/F/F'F1F/*&F'F/-% $sinG6#*&F/F/F'F1F/*&,&F/F/F'F1F/,&F/F/F'F/F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`lim B(t )` := map(limit, B(t), t=infinity) : Limit('B(t)', t=infinity) = mat rix(`lim B(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"B G6#%\"tG/F*%)infinityG-%'matrixG6#7$7%\"\"\"!#D#\"\"#\"\"$7%#F6F5F2!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 9 "Example 3" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "limit(E( t),t = infinity)" "6#-%&limitG6$-%\"EG6#%\"tG/F)%)infinityG" }{TEXT -1 9 " for a " }{TEXT 328 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 295 3 " \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 329 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 296 1 "E" }{TEXT 316 1 "(" }{XPPEDIT 18 0 "t " "6#%\"tG" }{TEXT 315 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "e[11](t) := sqrt(1/t+10) - sqrt(1/ t-81) : e[12](t) := (1-t^3)/t^2 : e[21](t) := (1/t +1)^2/sin(1/(t- 1)) : e[22](t) := sqrt(sin(1/t)+4) - sqrt(sin(1/t)-3) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "E(t) := matrix([ [e[11](t), e[12](t )], [e[21](t), e[22](t)]]) : 'E(t)' = E(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"EG6#%\"tG-%'matrixG6#7$7$,&*$-%%sqrtG6#,&*&\"\"\"F 4F'!\"\"F4\"#5F4F4F4*$-F06#,&F3F4\"#\")F5F4F5*&,&F4F4*$)F'\"\"$F4F5F4* $)F'\"\"#F4F57$*&*$),&F4F4F3F4FCF4F4-%$sinG6#*&F4F4,&F'F4F4F5F5F5,&*$- F06#,&-FJ6#F3F4\"\"%F4F4F4*$-F06#,&FSF4F@F5F4F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`lim E (t)` := map(limit, E(t), t=infinity) : Limit('E(t)', t=infinity) = m atrix(`lim E(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-% \"EG6#%\"tG/F*%)infinityG-%'matrixG6#7$7$,&*$-%%sqrtG6#\"#5\"\"\"F8^#! \"*F8,$F,!\"\"7$F,,&\"\"#F8*&^#F " 0 "" {MPLTEXT 1 0 22 "F(t) := matrix(2, 2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "F(t)[1,1] := (t-1 )*ln(t^2)/t^2 : F(t)[1,2] := ln(3+2^t)/(4*t+1) : F(t)[2,1] := ln(3 *t+1)-ln(2*t+5) : F(t)[2,2] := ln(1+2*exp(t))/t :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F(t) := F(t) : 'F(t)' = F(t) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"tG-%'matrixG6#7$7$*&*&,&F' \"\"\"F0!\"\"F0-%#lnG6#*$)F'\"\"#F0F0F0*$F6F0F1*&-F36#,&\"\"$F0)F7F'F0 F0,&F'\"\"%F0F0F17$,&-F36#,&F'F=F0F0F0-F36#,&F'F7\"\"&F0F1*&-F36#,&F0F 0*&F7F0-%$expGF&F0F0F0F'F1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`lim F(t)` := map(limit, F( t), t=infinity) : Limit('F(t)', t=infinity) = matrix(`lim F(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"FG6#%\"tG/F*%)infini tyG-%'matrixG6#7$7$\"\"!,$-%#lnG6#\"\"##\"\"\"\"\"%7$,&-F56#\"\"$F9F4! \"\"F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "`lim F(t)` := map(combine, `lim F(t)`) : Limit('F(t)', t=infinity) = mat rix(`lim F(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"F G6#%\"tG/F*%)infinityG-%'matrixG6#7$7$\"\"!,$-%#lnG6#\"\"##\"\"\"\"\"% 7$-F56##\"\"$F7F9" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-point evaluation yields" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`lim F(t)` := evalf(matrix(` lim F(t)`)) : Limit('F(t)', t=infinity) = matrix(`lim F(t)`) ;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"FG6#%\"tG/F*%)infinity G-%'matrixG6#7$7$$\"\"!F3$\"+_z'Gt\"!#57$$\"+\"3^Y0%F6$\"\"\"F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 348 9 "Example 5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "limit(G(t),t=in finity)" "6#-%&limitG6$-%\"GG6#%\"tG/F)%)infinityG" }{TEXT -1 9 " for a " }{TEXT 353 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 349 3 " \+ \327 " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 354 1 ")" }{TEXT -1 10 " m atrix [" }{TEXT 350 1 "G" }{TEXT 352 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT 351 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "G(t) := matrix(2, 2) :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 157 "G(t)[1,1] := (2*t^2+4*t-3)/(5*t^2-6*t+1) : G(t)[ 1,2] := (t^3-2)/(2*t^3+3*t-4) : G(t)[2,1] := (5*t+3)/(2*t-7) : G(t )[2,2] := (3*t^2+5*t-4)/(5*t^2-t+7) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "G(t) := G(t) : 'G(t)' = G(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"GG6#%\"tG-%'matrixG6#7$7$*&,(*$)F'\"\"#\"\"\"F1*& \"\"%F2F'F2F2\"\"$!\"\"F2,(F/\"\"&*&\"\"'F2F'F2F6F2F2F6*&,&*$)F'F5F2F2 F1F6F2,(F=F1*&F5F2F'F2F2F4F6F67$*&,&F'F8F5F2F2,&F'F1\"\"(F6F6*&,(F/F5* &F8F2F'F2F2F4F6F2,(F/F8F'F6FEF2F6" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "`lim G(t)` := map(li mit, G(t), t=infinity) : Limit('G(t)', t=infinity) = matrix(`lim G(t )`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"GG6#%\"tG/F*% )infinityG-%'matrixG6#7$7$#\"\"#\"\"&#\"\"\"F37$#F4F3#\"\"$F4" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 376 5 "* * *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 82 "Examples of limits of matrices containi ng functions as their variable approaches " }{TEXT 298 4 "zero" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 299 9 "Example 1" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "Limit(A(t),t=0) " "6#-%&LimitG6$-%\"AG6#%\"tG/F)\"\"!" }{TEXT -1 9 " for a " }{TEXT 330 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 300 3 " \327 " } {XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 331 1 ")" }{TEXT -1 10 " matrix [ " }{TEXT 301 1 "A" }{TEXT 318 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT 317 1 ")" }{TEXT -1 10 "] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "a[11](t) := 2*(sin(t/4))^3/t^3 : a[12](t) := (t-si n(t))^2/t^3 : a[13](t) := (1-cos(2*t))/t^2 : a[21](t) := ((t^2-t+4 )/(t^2-t+1))^(sin(2*t)/t) : a[22](t) := (sqrt(1+t)-1)/t : a[23](t) := ((1-cos(3*t))/t^2)^(sin(2*t)/t) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A(t) := matrix([ [a[11](t), a[12](t), a[13](t)], [a[ 21](t), a[22](t), a[23](t)] ]) : 'A(t)' = A(t) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"AG6#%\"tG-%'matrixG6#7$7%,$*&*$)-%$sinG6#,$F'#\" \"\"\"\"%\"\"$F6F6*$)F'F8F6!\"\"\"\"#*&*$),&F'F6-F2F&F;F " 0 "" {MPLTEXT 1 0 81 "`lim A(t)` := map(limit, A(t ), t=0) : Limit('A(t)', t=0) = matrix(`lim A(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"AG6#%\"tG/F*\"\"!-%'matrixG6#7$7%# \"\"\"\"#KF,\"\"#7%\"#;#F3F5#\"#\")\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 302 9 "Example 2" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "Limit(B(t),t=0)" "6#-%&LimitG6$-%\"BG 6#%\"tG/F)\"\"!" }{TEXT -1 9 " for a " }{TEXT 332 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 303 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 333 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 304 1 "B" }{TEXT 320 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 319 1 ")" }{TEXT -1 10 " ] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "b[11](t) := \+ ln(sin(2*t))/ln(sin(t)) : b[12](t) := sin(2*t)/t : b[13](t) := t^s in(t) : b[21](t) := t^t : b[22](t) := t/sin(3*t) : b[23](t) := ( sin(t))^t :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "B(t) := mat rix([ [b[11](t), b[12](t), b[13](t)], [b[21](t), b[22](t), b[23](t)]]) : 'B(t)' = B(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"BG6#%\"tG -%'matrixG6#7$7%*&-%#lnG6#-%$sinG6#,$F'\"\"#\"\"\"-F/6#-F2F&!\"\"*&F1F 6F'F:)F'F97%)F'F'*&F'F6-F26#,$F'\"\"$F:)F9F'" }}}{EXCHG {PARA 2 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "`lim B(t) ` := map(limit, B(t), t=0) : Limit('B(t)', t=0) = matrix(`lim B(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"BG6#%\"tG/F*\"\" !-%'matrixG6#7$7%\"\"\"\"\"#F27%F2#F2\"\"$F2" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 355 9 "Example 3" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "limit(E(t),t=0)" "6#-%&limitG6$-%\"EG 6#%\"tG/F)\"\"!" }{TEXT -1 9 " for a " }{TEXT 360 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 356 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 361 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 357 1 "E" }{TEXT 359 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 358 1 ")" }{TEXT -1 10 " ] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "E(t) := matri x(2, 3) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "E(t)[1,1] := \+ (tan(t)-t)/t^3 : E(t)[1,2] := (1-2*sin(t)^2-cos(t)^3)/(5*t^2) : E( t)[1,3] := sinh(t)/t : E(t)[2,1] := sin(t)^2/t^2 : E(t)[2,2] := (s inh(t)-t)/t^3 : E(t)[2,3] := (tan(t)*arctan(t)-t^2)/t^6 :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "E(t) := E(t) : 'E(t)' = E( t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"EG6#%\"tG-%'matrixG6#7$7% *&,&-%$tanGF&\"\"\"F'!\"\"F1*$)F'\"\"$F1F2,$*&,(F1F1*&\"\"#F1)-%$sinGF &F:F1F2*$)-%$cosGF&F5F1F2F1*$)F'F:F1F2#F1\"\"&*&-%%sinhGF&F1F'F27%*&*$ F;F1F1*$FCF1F2*&,&FGF1F'F2F1*$F4F1F2*&,&*&F/F1-%'arctanGF&F1F1*$FCF1F2 F1*$)F'\"\"'F1F2" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "`lim E(t)` := map(limit, E(t), t=0) : Limit('E(t)', t=0) = matrix(`lim E(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"EG6#%\"tG/F*\"\"!-%'matrixG6#7$7%#\"\" \"\"\"$#!\"\"\"#5F37%F3#F3\"\"'#\"\"#\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 362 9 "Example 4" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "limit(F(t),t=0)" "6#-%&limitG6$-%\"FG 6#%\"tG/F)\"\"!" }{TEXT -1 9 " for a " }{TEXT 366 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 363 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 367 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 365 1 "F" }{TEXT 368 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 364 1 ")" }{TEXT -1 10 " ] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "F(t) := matri x(2, 3) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "F(t)[1,1] := \+ (t-sin(t))/(t-tan(t)) : F(t)[1,2] := (sin(t)-t*cos(t))/t^3 : F(t)[ 1,3] :=t^3/(sin(t)-t) : F(t)[2,1] := (tan(t)-sin(t))/t^3 : F(t)[2, 2] := (tan(t)-t)/(t-sin(t)) : F(t)[2,3] := (exp(t)+exp(-t)-2)/(2*cos (2*t)-2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F(t) := F(t) \+ : 'F(t)' = F(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"FG6#%\"tG- %'matrixG6#7$7%*&,&F'\"\"\"-%$sinGF&!\"\"F/,&F'F/-%$tanGF&F2F2*&,&F0F/ *&F'F/-%$cosGF&F/F2F/*$)F'\"\"$F/F2*&*$F " 0 "" {MPLTEXT 1 0 81 "`lim F(t)` := map(limit, F(t), t=0) : Limit('F(t)', t=0) = matrix(`lim F(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"FG6#%\"tG/F*\"\"!-%'matrixG6#7$7%#!\"\" \"\"##\"\"\"\"\"$!\"'7%#F6F4F4#F3\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 369 9 "Example 5" }}} {EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Compute " }{XPPEDIT 18 0 "limit(G(t),t=0)" "6#-%&limitG6$-%\"GG 6#%\"tG/F)\"\"!" }{TEXT -1 9 " for a " }{TEXT 373 1 "(" }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT 370 3 " \327 " }{XPPEDIT 18 0 "3" "6#\"\"$" }{TEXT 374 1 ")" }{TEXT -1 10 " matrix [" }{TEXT 372 1 "G" }{TEXT 375 1 "(" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 371 1 ")" }{TEXT -1 10 " ] given as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "G(t) := matri x(2, 3) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "G(t)[1,1] := \+ (sinh(t)-tanh(t))/t^3 : G(t)[1,2] := (t+sin(t))/(t^2+t) : G(t)[1,3 ] := (ln(3*sin(t))-ln((1+t)*sin(t)))/(2*exp(t)-1) : G(t)[2,1] := (co t(t)-1/t)/(coth(t)-1/t) : G(t)[2,2] := (exp(sin(t))-t-1)/t^2 : G(t )[2,3] := (4^t-2^t-t*(ln(4)-ln(2)))/t^2 :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "G(t) := G(t) : 'G(t)' = G(t) ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"GG6#%\"tG-%'matrixG6#7$7%*&,&-%%sinhGF&\"\"\"-%% tanhGF&!\"\"F1*$)F'\"\"$F1F4*&,&F'F1-%$sinGF&F1F1,&*$)F'\"\"#F1F1F'F1F 4*&,&-%#lnG6#,$F:F7F1-FC6#*&,&F1F1F'F1F1F:F1F4F1,&-%$expGF&F?F1F4F47%* &,&-%$cotGF&F1*&F1F1F'F4F4F1,&-%%cothGF&F1FRF4F4*&,(-FL6#F:F1F'F4F1F4F 1*$F>F1F4*&,()\"\"%F'F1)F?F'F4*&F'F1,&-FC6#FhnF1-FC6#F?F4F1F4F1*$F>F1F 4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 "`lim G(t)` := map(limit, G(t), t=0) : Limit('G(t) ', t=0) = matrix(`lim G(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%& LimitG6$-%\"GG6#%\"tG/F*\"\"!-%'matrixG6#7$7%#\"\"\"\"\"#F4-%#lnG6#\" \"$7%!\"\"F2,$*$)-F66#F4F4F3#F8F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Floating-point evaluation \+ yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`lim G(t)` := eva lf(matrix(`lim G(t)`)) : Limit('G(t)', t=0) = matrix(`lim G(t)`) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$-%\"GG6#%\"tG/F*\"\"!-%' matrixG6#7$7%$\"+++++]!#5$\"\"#F,$\"+*G7')4\"!\"*7%$!\"\"F,F2$\"+5_z1s F4" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 " " {TEXT 377 5 "* * *" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Proceed to Unit (29) for \"" } {TEXT 340 44 "Integration of matrices comprising functions" }{TEXT -1 2 "\"." }}}{EXCHG {PARA 259 "" 0 "" {TEXT 339 67 "-------------------- -----------------------------------------------" }}}}{MARK "29 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }