(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 119522, 2604] NotebookOptionsPosition[ 111253, 2397] NotebookOutlinePosition[ 111748, 2417] CellTagsIndexPosition[ 111705, 2414] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["Limits\n", "Title"], "\nToday we will be looking at limits using Mathematica. Mathematica is a \ great tool where we can graph and find the limits of functions. First start \ by opening up Mathematica on your computers. Go to \ \[OpenCurlyDoubleQuote]Search Windows\[CloseCurlyDoubleQuote] on the bottom \ left corner of your desktop. Start Typing in \ \[OpenCurlyDoubleQuote]Mathematica\[CloseCurlyDoubleQuote] It should pop up \ and you can click on it to open the program.\n\tGreat now to start your going \ to click this button ", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzt0b1uglAAxXFSl46+gm/h0DTp6KrxASSlpgtNsInx7QgLXxMjYWaChZDA AtTjUKNeVExEMZ5fgtHLfzi5DuSf8deLJEmLV3yMZ8sPTZutJn38mKqL77mq fI7UX2WuaEO5h8N3PG94Nt//iIiIiIiIiP7pup7nee25eIhSv63azZZlXbQZ /YW3cn2PuNk0zd3Np/8glOjvMXPPweatY/fckc1ZlonntZtRdmGz67pJkojn tZtRom9/1Bm+74dh2DBGib7VPU2kaeo4TsMYJfpW9zRRVRWuLgiCsyUalOhv sOo0bCjLEmM8z4uiqCgKscE53qJB2YXNW3Ec27ZtGIYlwDne3nsgERERERER ERER0VNYA/hXP0s= "], {{0, 62}, {60, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSize->Automatic, ImageSizeRaw->{60, 62}, PlotRange->{{0, 60}, {0, 62}}]], "Input",ExpressionUUID-> "60138429-3163-45c6-80f4-3717891f70ed"], " Then you are going to select \[OpenCurlyDoubleQuote]Plain Text\ \[CloseCurlyDoubleQuote]. Then you can start typing so we are going to call \ this \[OpenCurlyDoubleQuote]Limits\[CloseCurlyDoubleQuote] So go ahead and \ type that. 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This will change your font and color of your words to look more like a \ title of the document. Now click on \[OpenCurlyDoubleQuote]Palettes\ \[CloseCurlyDoubleQuote] and Click \[OpenCurlyDoubleQuote]Classroom Assistant\ \[CloseCurlyDoubleQuote] this tool box will help us throughout today\ \[CloseCurlyQuote]s lesson.\n\tWe know in mathematics, a limit is the value \ that a function or sequence \[OpenCurlyDoubleQuote]approaches\ \[CloseCurlyDoubleQuote] as the input or index approaches some value. So lets \ input some examples. To in put an expression you will need to type \ Limit[(your function), x\[Rule]0] so lets try this function. Your arrow can \ be found on the classroom assistant.\n\tLimit[Sin[x]/x,x\[Rule]0] then press \ Shift Enter. 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If you would like your function to look like this ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"Sin", "[", "x", "]"}], SuperscriptBox["x", "2"]], TraditionalForm]],ExpressionUUID-> "a860539e-1d0a-4613-a489-1f7bda0346f6"], " you can use the classroom assistant. Next you can first define your \ function then calculate the limit. 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Then to use this function to find the \ limit you will type ", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztV71OwzAQrmBhZOAFeAtUZWFkLeIBUpFULEFKkRBDp6p7KvUR2qUPULFW bOydMnVF6jMU2/nBse+CEyekSPdJaePz6Xz3ff7L9fB54J/1er3xBfsZuK+3 Yei+3V+yxkMwfhoF3uNd8OKNvPBmeM6MV+yZsYe/HwkEAoFAaA8bn501/Shu 0vOXGAL+xibOP0cc9REqreQQYSFiM9oLfVgSoDOWgP2UqA9RgJpDnaxwOezz Q+TQzSJxwBt0xsCcO5JDSl+rpGJWoBw/Yuc9PKzvc3P6lwwKexZRiG4jx8d6 7kzm7vbAG7t39u6sY9m5khzl0QDEUQTNDZnBlA4p6SbkAHpSxjIjb+aDAjFM V4e0yo20229XzmS1/IqnGns1VkdJNDBzcNLkc4nX4kcKFy3KwcbkRm5rSg6l IgPnw3LBZ7Wz+NxrztU3KzQaDCjRxMZlyDQhOY5I4cnqwwkpk0NbueAKVjYo tZQTkgMKaykH3/A5daabVXqQIYyURAOA1STZdZc6ckgQ5aunMTMicgCeRSKa P8qnO9EwPcrR+3p5NB1lxEo06FtZZ9dvA1S9u9pedFu6yBvj9OWAJpGdM/LB hd2F/gidf5wSCAQCgUBoGd+qBEGz "], {{0, 20}, {133, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSize->Automatic, ImageSizeRaw->{133, 20}, PlotRange->{{0, 133}, {0, 20}}]], "Input",ExpressionUUID-> "878fbe1b-eed7-4256-a312-4b639b72ac29"], "then press Shift Enter again and you should get 0. \n\tSome functions have \ no definite limit. For these the output will be an interval, which is where \ the uncertain value lies between the interval of ( x-min, x-max). For example \ type in this function and hit Shift Enter and you should get an interval \ response. 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Lets plot the function ", Cell[BoxData[ FormBox[ FractionBox["1", "x"], TraditionalForm]],ExpressionUUID-> "dc03bf9f-2d30-4b13-ab51-0e0f1e015959"], " to do this you can click Basic commands on your classroom assistant then \ 2D the Plot and you can input the information. 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And if \ we want to do from the other direction we can type", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztnDtu20AQhomkSZkr5BaJoCZlWgc5gIyYRhoFkAMEKVz5AjTgI9iNDyCk NdKld6UmbgP4DM7yPVzOrEhLpijn+wDZ3OU+Zmd+7S4JrN4cfj2IX0RRdPLK /TmYfX+/WMx+fHztEp/mJ1+O50efP8y/HR0fLd4evnSZ79znj/uk1w8AAAAA AAAAAAAAAAAAfVnGURRNktU2S4LFYD7cs2Ctkolh7kb6zJqNl0bZyL+X52rl ZYlxeTUbYk5Hw0SNVqXty8YIwfD6bAe3FEEHM2x9bkRIn152aWxQn67QuNSZ WZ0bLC7DSF8/vUzMEAxJILjdQqrqs/6iV3fS1uI4zS7+5R3qJZs0Wvf1mTbg MsIxLgpJfl2fT0/PZzf3aeL2p7ueXq8Pdu9aqyTRjJJeK1zQQQkNX9cJxYd5 pkuW98rW9UlbTsxpthECtaNGwOSQ4rjMXzs0w0vh4G6gT+VO0UeZmSarHpU2 esyfsnndRmMkdzdX09Ory7+rs27ifEQta99RmZtFIOm4BqkeVe8V+nQScTn1 LbHJqWvnqmvbaITA66g1p5ejkpdrxrZudzaUPl0Xxdf6YUB9ll0q3F9epPPh 9OL3XWiMm9XSTMvzkmJa77pH6qtPfZfuTYtm55302SxU6l8VbZhQsWetT2Vt r9CVlofWHntIn81HGGuT4q3pDeMDvRvru5ZUHag6aRf67OiloN0j0KfWbF99 BkaRbiZTjfkrdTHNGNWsWjrWOES+VyTQeyjsXfSpr+TB9b1thmauP5C+8+ea BWRjfQqyZvyFpNjjKvpUSkqj2qHSno/sr6C9tudPOme3WcJ/0jEfjoO1FEI+ FENXlmFTn+1Kqg9Ds1/LVbIF0bEfAj1Yssmiw576NL0UDO7oXslkGPOnWfgx b0ie6J3ZXvS+P4xWn8qUs+32d/Xab7e97w093s8DAAAAAAAAAAAAAAAAAAAA AAAAAAAAAPznVMepJsmyOujE2QUYFO9QnzgZndTnh+P6PGrCmREYAUKIUp/W T+0ADIs4vJz/whBH7gAAAAAAAAAAAAAAAAAAAAAAAOB58w891bkj "], {{0, 57}, {224, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSize->Automatic, ImageSizeRaw->{224, 57}, PlotRange->{{0, 224}, {0, 57}}]], "Input",ExpressionUUID-> "2d3f5581-3b45-4647-9a1f-9429ed9d1b18"], "(limit as it approaches 0 from below) we see that the output is negative \ infinity. \n\tLet\[CloseCurlyQuote]s define the function as f[x] so first \ delete the previous f[x] we defined above. 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This is where we would need to find the limit. We type in ", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztmz1M20AYhq12yZiRke4MDB2qKAPd6IaokBgDKi5daASVqgohBgbWILFk QFlggYEt6oCEEANShzChDJFCmZAqfibEmH5nx7+5s892/BPnfaQk9p1zuThP vnzny71b/j6vvlEUZaNAd/NLPz+ury/9+lyknYW1jW9f11a+fFr7sfJ1Zf3D 8lsqfE+3B7qx7R4AAAAAAAAAAAAAAABkkkajoaRKpVJJ+xyA0WZxcbFarZ6n R7vdTvscgBHm9fW1WCzCIjC6UBicnJxMuxcAhKeqkXYvAJDl/v7+9vbWXkJB mEKxraBTK1mjLbUZRy+Ml4indZBXyF6Kt4VCYW5uziykNJhKKCXmPUMzrVTr DLsnerulEhweOy6P98pbe0sXj2zn5jdtl48D+DU7O1upVK6urkja6+trvXB3 d9eutBOOw5J9aKpi97VW1Wb/Qb7/IB/cXRyVt44O/3V2Agpsh6Kx6e3MzEyj 0XAfYWUTHMlk+iB22FQXDuccckDg0OPhPouE5f0/d2EbpyBMLdP98/MzbVCO ITpSkEz490HoMHtjek2mHTa/xDFkUuMCfdSCj9fLH9N83/NPcXh1dZUi8PT0 tE8/An6PPPvgGC369zJtvDIi4IfIYZaOMnMi5RI9LRRTVkyJxPb2tnc3BhWW 6YPEp5/pOKwDh6PAdVgfT+3caDvBx3QuKBSToDTEc1d4XlqT7AMcBuJcYmhQ KJ6YmIj3NUaccXFY+7mVeqfyRybiMHF6ehr7a4wy+XJYfLU/ksPC39NkHAYG /HnDcXE4erNwOG1E84axOdw9O2k98Yrr9bpZ89Q6Oeuam3WGtq/vGFXycB22 Bj9mDXvXqsqK+w/6WeEf6XGNCQ57E3jKslOric6neN4weYfNYk1nQ1Sj3Ky3 6S2POA47a/oXo4xCtmueFk4biMPhCTRlqUcM3in1mjeM3WHdVNNWy+Futzso qlVgbPWfYITpuj2MDxDEYToPrJCVDddhzhqefOH6Q50fAacsuRe7PecNk3DY SA/YozM8Ox3WdLflFvHG4dgcTnGtUDK4/j6nZ2Bii7wc5s0Iajia85k3TMBh /VHCYcfzXFXpx2Fes8gleuaQQaBRsClL/0F5OnFYxmHLWIHDcri/sNp7dQ/L qFDgMOfIPlYNxnSDCC9cBp2yDDdvmBGHrUBrVLvy4WwChxlx/QdflqSvD0sr GSoOJw0cFl9JSLQPCTvsmdD2CXt9OGngcBbI1zxd0iTiMBZm+gCHo5CAw1iY 6QscjkLsDkdYmNncvFSmLksHL2znvE3byuZDLJ1MGzgchZgdjrows3PQUqZa tb8Pan4F7sHhaIjXhA6r9YgLM19qCywaKwt3ufyMsSY02/gvzJRYW5pzh8Ho EDIOs5SY2ZvzXAKMAuHHdOq5tpPrMR0AAAAAAAAAAAAAAAAAAAAAAACQIv8B c/Mx+Q== "], {{0, 52}, {236, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSize->Automatic, ImageSizeRaw->{236, 52}, PlotRange->{{0, 236}, {0, 52}}]], "Input",ExpressionUUID-> "4dc643a1-4ff2-490e-993f-00c76de707bd"], "then press Shift Enter and you should get ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"-", "5"}], "8"], TraditionalForm]],ExpressionUUID-> "b2a2d93f-5d77-47bb-b9db-5bab88553f9e"], ". To get more of a visual on this function lets Plot it. \n", Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzt3V2LHFd+B2Ale5OrkK+Qb7EsvsllbjfkdsGreNa5sAPagAl4LxpHLCY2 SLBrGJIOXrAu1sKGsGhkG2mxLRxtJOFGiLWMR4wzEZsxsleyI/yCjJmcmdNd fbqquqa7p7un68zzUJZrTnXXS1f1r//11v2XP/6HH6796YkTJ376Z+GfHz7+ zF+dOvX4P/3NX4Q//vbpn/79T55+4u/++ul/fOInT5z6/o+/Fxr//E9OnPi3 8N9e/y4AAAAAAAAAAAAAAAAA87F55gcnoh+c2Zz72DfW+iM/sbYx95EzP4vd DGAhhpttZdPdHzTBxrz/uLp0GmTXyLBxY619cI3B42Z8l/WfvuAwjVOZdiLV ZyVrZ7i8TY2zZs8iNoPRca7ox9e028Oc12zldT3k5s3u6Ba7/3oOX/h5ZFq1 uTSR5gePEx4820qfaiqzma0KrXtW8lINe2sbh0+Y8b2wiM0gaRi30o/ctFvd nNbscFDtCzv7elw5W5fOv3+/rnl9fX1/yP33z6+vD/6IQtOlrX5PGBL7p5Bu saWtt7oxD1dPvz1pqPl4OUymXXn17GM/O/v4O3/c++P3b4T+x17dTB88l0xr nsos9l60MIUp38R1z6qsmb1BtY3p0s0h0+a1GSSZloxjpHpLp7i2NhhLsUSl 8SbjmrSY2TxzpnElTJFp81uz6dSPbabF5jS+Ys9+2g1zrGifQmVrG/sBPfaj fNI6bXRf5OAA3H7n3GM/O/fKvc3nKlEzxzqtYSqHMFthMvKs4o/9t8WZ+Oqf qWvcTJ5/+Eyb22YwbCiPsjymJJ2T0e+NPZmlYcVXX0/WOXBPcfq6fQ5rNm09 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You can \ play around with it and input 10, 5 or other numbers.\n\t \n\tWhen typing a \ graph and you need to change the y axis you need to type PlotRange\[Rule] \ then you y min and max. 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