0001000101000feActProject.ACT000102000aLimits.EAC0100000050f3ÉÐ<Ì$W[Limits[\AuthoráàœBarry Kissane School of Education Murdoch University Murdoch, WA, Australia kissane@murdoch.edu.au http//wwwstaff.murdoch. edu.au/~kissane ÿ[[ 1. Objective:[To explore an important[circular function limit[from first principles.[[ 2. Example:[Consider the function[f(x)=sin(x)x . We will[explore this function for[values of x close to 0.[\Tap to see graph -î†ÌʨGraph2DT †Graph3Dh †LISTSYSt4N†Modify ¨$ †STATCALC ÌN†STATSYS Ô\N†Sequence0, †Sheet\| †Sheet3DØ| †SolveEqT†SolveLwrX †SolveUprd †StupFLG1p(†StupList˜D †StupPictÜ$ †ViewWind †y1H†ä8 †äD †äP †ä\ †äh †ät †ä€ †äŒ †ä˜ †ä¤ †ä° †ä¼ †äÈ †äÔ †äà †äì †ä ø †ä! †ä" †ä# †ä$( †ä%4 †ä&@ †ä'L †ä(X †ä)d †ä*p †ä+| †ä,ˆ †ä-” †ä.  †ä/¬¤ †ä0P †ä1\ †ä2h †ä3t †ä4€ †ä5Œ †äE˜ †äF¤ †äH° †äI¼ †äJÈ †äKÔ †äLà †äMì †äNø †äO †äP †äQ †äR( †äS4 †äT@ †ä]L †ä^P †ä_T †ä`X †äa\ †äb` †ä”d †ä•p †äÍ| †äΈ †äД †systemä]ä^systemä^systemä_ä`systemä`äasystemäaäbsystemäb( LISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLIST abGrapbGraph2D systemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLIST  0@PSheet1Œ ýSheet2Œ ýSheet3Œ ýSheet4Œ ýSheet5Œ ýSheetr-value:í 0@PSheet1¸Œ û8Œ û Sheet2¸Œ û8Œ û Sheet3¸Œ û8Œ û Sheet4¸Œ û8Œ û Sheet5¸Œ û8Œ û Sheetr-value:í i™ ™Ð<Ì$ $   3xàL(x)/x`P`P` BGw–i@` BGw–i@W–2g”"9—wb& ™``` ™(1…0qy`#Y‡uYƒ ˜` ™ $0<HT`lx„œ¨´ÀÌØäðü ,8DP\ht€Œ˜¤°¼ÈÔàì`AG „€x• ™ Y™p622‘”$ ™Y™–iQbD ™Y™ 1 v) ™Y™ Aˆ™% ™Y™ X…w „ ™Y™ sTXUw% ™Y™ …sUSw– ™Y™ “4fS—S ™Y™ ˜3AfF‚y ™€Ð – ™™™˜331Ð ™ ™ “4fS—S ™ ™ …sUSw– ™ ™ sTXUw% ™ ™ X…w „ ™ ™ Aˆ™% ™ ™ 1 v) ™ ™–iQbD ™ ™p622‘”$ ™AG „€x• ™€`!&1W‰G ˜Q[[Trace (Analysis menu) [left and right to see the[function is not defined[for x = 0, but is [defined for all other[ values of x.[[Close the graph from the[î  menu when you have[ studied it. [[A table of values may[help to see what [happens when x is very[ close to 0.[\Tap to see table -î†ÌʨGraph2DT †Graph3Dh †LISTSYSt4N†Modify ¨$ †STATCALC ÌN†STATSYS Ô\N†Sequence0, †Sheet\| †Sheet3DØ| †SolveEqT†SolveLwrX †SolveUprd †StupFLG1p(†StupList˜D †StupPictÜ$ †ViewWind †y1H†ä8 †äD †äP †ä\ †äh †ät †ä€ †äŒ †ä˜ †ä¤ †ä° †ä¼ †äÈ †äÔ †äà †äì †ä ø †ä! †ä" †ä# †ä$( †ä%4 †ä&@ †ä'L †ä(X †ä)d †ä*p †ä+| †ä,ˆ †ä-” †ä.  †ä/¬¤ †ä0P †ä1\ †ä2h †ä3t †ä4€ †ä5Œ †äE˜ †äF¤ †äH° †äI¼ †äJÈ †äKÔ †äLà †äMì †äNø †äO †äP †äQ †äR( †äS4 †äT@ †ä]L †ä^P †ä_T †ä`X †äa\ †äb` †ä”d †ä•p †äÍ| †äΈ †äД †systemä]ä^systemä^systemä_ä`systemä`äasystemäaäbsystemäb( LISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLIST abGrapbGraph2D systemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLIST  0@PSheet1Œ ýSheet2Œ ýSheet3Œ ýSheet4Œ ýSheet5Œ ýSheetr-value:í 0@PSheet1¸Œ û8Œ û Sheet2¸Œ û8Œ û Sheet3¸Œ û8Œ û Sheet4¸Œ û8Œ û Sheet5¸Œ û8Œ û Sheetr-value:í i™ ™Ð<Ì$ $   3xàL(x)/x`P`P` BGw–i@` BGw–i@W–2g”"9—wb& ™``` ™(1…0qy`#Y‡uYƒ ˜`  ™ $0<HT`lx„œ¨´ÀÌØäðü ,8DP\ht€Œ˜¤°¼ÈÔàì`AG „€x• ™ Y™p622‘”$ ™Y™–iQbD ™Y™ 1 v) ™Y™ Aˆ™% ™Y™ X…w „ ™Y™ sTXUw% ™Y™ …sUSw– ™Y™ “4fS—S ™Y™ ˜3AfF‚y ™€Ð ˜ ™˜33AfdÐ ™ ™ “4fS—S ™ ™ …sUSw– ™ ™ sTXUw% ™ ™ X…w „ ™ ™ Aˆ™% ™ ™ 1 v) ™ ™–iQbD ™ ™p622‘”$ ™AG „€x• ™Ð ˜ ™˜33AfdÐ ™`!&1W‰G ˜Q[[After completing the next[step, select î  eActivity[for more instructions.[[Tap in graph window and[select table icon on the[left end of the toolbar.[Scroll the table to see[values.[[Highlight a value in the x[column and change it to[a value closer to 0.[Observe the new y-value[[Repeat several times,[with values increasingly[ close to 0.[[Close the table from the[î  menu when you have[ studied it. [[**Study limits [ numericallyR define f(x)=sin(x)xRdoneR define F(x)=listToMat(f(matToList(x,1))) RdoneRF0.10.010.001-0.1-0.01-0.001R 0.9983341665 0.9999833334 0.9999998333 0.9983341665 0.9999833334 0.9999998333R[ 3. Exercise[[Explore the limit of[ 1-cos(x)x [as x approaches 0, in a[similar way, using both [graphs and tables.[[Solutions to Example[ and Exercise:[Limits can be formally[evaluated from the 2D[keyboard or from Main[(using Calculation[menu). The link below[shows both methods.[[\Limit results -----î†î†Ë߀R sin(í¸)í¸í¸0R1Rlim((1-cos(í¸))/í¸,í¸,0,0)R0RÀGraph2D, †Graph3D@ †LISTSYSL4N†Modify €$ †STATCALC ¤N†STATSYS ¬\N†Sequence, †Sheet4| †Sheet3D°| †SolveEq,†SolveLwr0 †SolveUpr< †StupFLG1H(†StupListpD †StupPict´$ †ViewWindØ †äô †ä †ä †ä †ä$ †ä0 †ä< †äH †äT †ä` †äl †äx †ä„ †ä †äœ †ä¨ †ä ´ †ä!À †ä"Ì †ä#Ø †ä$ä †ä%ð †ä&ü †ä' †ä( †ä) †ä*, †ä+8 †ä,D †ä-P †ä.\ †ä0h †ä1t †ä2€ †ä3Œ †ä4˜ †ä5¤ †äE° †äF¼ †äHÈ †äIÔ †äJà †äKì †äLø †äM †äN †äO †äP( †äQ4 †äR@ †äSL †äTX †ä]d †ä^h †ä_l †ä`p †äat †äbx †ä”| †ä•ˆ †äÍ” †äΠ †äЬ †systemä]ä^systemä^systemä_ä`systemä`äasystemäaäbsystemäb( LISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLISTLIST abGrapbGraph2D systemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLISTsystemä]ä^systemä^systemLIST  0@PSheet1Œ ýSheet2Œ ýSheet3Œ ýSheet4Œ ýSheet5Œ ýSheetr-value:í 0@PSheet1¸Œ û8Œ û Sheet2¸Œ û8Œ û Sheet3¸Œ û8Œ û Sheet4¸Œ û8Œ û Sheet5¸Œ û8Œ û Sheetr-value:í i™ ™Ð<Ì$ $ `P`P`p`p ™€`€(1…0qy`#Y‡uYƒ ˜` ™O[eAct FH$xà^(f(àa(x,1))) angles Ô  $0<HT`lx„Œœœœœœœœœ œ@œ`œu8Åarea D $0<HT`lx„œ¨´ÀÌØäY5Ya`HyTe`a€I’™(X6f“‘G‰H 3…U2F`7sC)E(8xGEaRB'HgC63@TG…QˆC'S8sP06C† 62‘„)GU2AX7 T X6b”3SfH x((àL(x))/(x))m_angle m_area volume D $0<HT`lx„œ¨´ÀÌØäY5Ya`HyTe`a€I’™(X6f“‘G‰H 3…U2F`7sC)E(8xGEaRB'HgC63@TG…QˆC'S8sP06C† 62‘„)GU2AX7 T X6b”3Sx D $0<HT`lx„œ¨´ÀÌØäœ —œ ˜œ ™¬ý     ( 2 < P d x Œ   ¯xœyœyœy