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极限练习：多类型函数极限的动态演示与计算

极限练习：多类型函数极限的动态演示与计算
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2020-08-20 11:15
函数三角函数微积分导数极限
① 本范例演示如何通过随机生成的多种函数（有理式、三角函数、分段函数、多项式等）动态计算并展示其极限，包括左右极限与双侧极限。② 涉及的核心数学概念有极限的定义与性质、一侧极限、连续性判断、因式分解与有理化技巧、三角函数极限公式以及分段函数的极限处理。③ 作图要点：利用 RandomDiscrete、RandomElement、Random、If 等命令随机生成题目参数；用 Limit、LimitBelow、LimitAbove 计算双侧与单侧极限；用 Expand、Factor、Element、DivisorsList、Factors 对表达式进行展开、因式分解与提取因子；通过 Textfield、FractionText、SurdText、TableText 实现交互式输入与结果显示；相关对象如 factoringnumerator、a_{trignum}、b_{trignum}、f_{factor}、f_{pwL1} 等承载具体函数部件。④ 适用于高中选修微积分、大学微积分入门课程的极限专题教学与自主练习，也可作为 GeoGebra 脚本教学的参考范例。

GeoGebra 命令及参数

  1. RandomDiscrete({1, 2}, {65, 35}) → factoringnumerator
  2. RandomElement({-4, -3, -2, -1, 1, 2, 3, 4}) → b_{2}
  3. Random(-12, 12) → a_{trignum}
  4. Random(-10, 10) → c
  5. Random(-12, 12) → b_{trignum}
  6. Random(-12, 12) → a_{1}
  7. Random(-4, 4) → m_{1}
  8. If(m_{1} ≠ 0, m_{1}, 1) → m
  9. Random(1, 12) → a_{pos}
  10. Random((-a_{pos}) + 1, 15) → a_{big}
  11. RandomElement({4, 6}) → a_{trigden}
  12. RandomDiscrete({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 2, 3, 1, 3}) → derwhatweget
  13. If(0 ≤ derwhatweget ≤ 6, a_{trig}, 7 ≤ derwhatweget ≤ 8, a_{big}, derwhatweget ≟ 16, a_{big}, c) → a_{der}
  14. Random(-10, 10) → b_{1}
  15. Random(2, 10) → logbase
  16. RandomElement({4, 6}) → b_{trigden}
  17. If(a_{1} ≠ 0, a_{1}, 3) → a
  18. If(b_{1} ≠ 0 ∧ b_{1} ≠ a, b_{1}, 11) → b
  19. RandomDiscrete({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0}) → factoringwhatweget
  20. If(factoringwhatweget ≟ 1, x - a_{pos}, factoringwhatweget ≟ 6, x - a_{pos}, factoringwhatweget ≟ 7, x^(3) - a_{pos}^(3), factoringwhatweget ≟ 8, x^(2) - a_{pos}^(2), factoringwhatweget ≟ 9, x - a_{pos}, factoringwhatweget ≟ 10, x + a_{pos}, factoringwhatweget ≟ 2, factoringwhatweget ≟ 11, x^(2) - a_{pos}^(2), factoringwhatweget ≟ 12, x^(2) - a_{pos}^(2), factoringwhatweget ≟ 13, x - a_{pos}^(2), factoringwhatweget ≟ 14, x - a_{pos}^(2), factoringwhatweget ≟ 15, sqrt(x) - a_{pos}, x - a_{pos}, factoringwhatweget ≟ 16, sqrt(x) + a_{pos}, factoringwhatweget ≟ 3, sqrt(x) - sqrt(a_{pos}), factoringwhatweget ≟ 4, sqrt(x) + sqrt(a_{pos}), factoringwhatweget ≟ 5, x^(3) - a_{pos}^(3)) → f_{factord}
  21. If(factoringwhatweget ≟ 1, sqrt(a_{pos}) - sqrt(x), factoringwhatweget ≟ 6, a_{pos}^(3) - x^(3), factoringwhatweget ≟ 7, a_{pos}^(2) - x^(2), factoringwhatweget ≟ 8, a_{pos}^(3) - x^(3), factoringwhatweget ≟ 9, a_{pos}^(2) - x^(2), factoringwhatweget ≟ 10, a_{pos}^(2) - x^(2), factoringwhatweget ≟ 2, factoringwhatweget ≟ 11, a_{pos} - x, factoringwhatweget ≟ 12, a_{pos} + x, factoringwhatweget ≟ 13, a_{pos} - sqrt(x), factoringwhatweget ≟ 14, a_{pos} + sqrt(x), factoringwhatweget ≟ 15, a_{pos}^(2) - sqrt(x), sqrt(a_{pos}) + sqrt(x), factoringwhatweget ≟ 16, a_{pos}^(2) - sqrt(x), factoringwhatweget ≟ 3, a_{pos} - x, factoringwhatweget ≟ 4, a_{pos} - x, factoringwhatweget ≟ 5, a_{pos} - x) → f_{factornopp}
  22. If(factoringwhatweget ≟ 1, sqrt(x) - sqrt(a_{pos}), factoringwhatweget ≟ 6, x^(3) - a_{pos}^(3), factoringwhatweget ≟ 7, x^(2) - a_{pos}^(2), factoringwhatweget ≟ 8, x^(3) - a_{pos}^(3), factoringwhatweget ≟ 9, x^(2) - a_{pos}^(2), factoringwhatweget ≟ 10, x^(2) - a_{pos}^(2), factoringwhatweget ≟ 2, factoringwhatweget ≟ 11, x - a_{pos}, factoringwhatweget ≟ 12, x + a_{pos}, factoringwhatweget ≟ 13, sqrt(x) - a_{pos}, factoringwhatweget ≟ 14, sqrt(x) + a_{pos}, factoringwhatweget ≟ 15, x - a_{pos}^(2), sqrt(x) + sqrt(a_{pos}), factoringwhatweget ≟ 16, x - a_{pos}^(2), factoringwhatweget ≟ 3, x - a_{pos}, factoringwhatweget ≟ 4, x - a_{pos}, factoringwhatweget ≟ 5, x - a_{pos}) → f_{factorn}
  23. If(factoringnumerator ≟ 1, Simplify[f_{factorn}] / Simplify[f_{factord}], factoringnumerator ≟ 2, Simplify[f_{factornopp}] / Simplify[f_{factord}]) → f_{factor}
  24. Limit(f_{factor}, a_{pos}) → FactorLimitA
  25. RandomElement({-4, -3, -2, -1, 1, 2, 3, 4, 1 / 2, -1 / 2, 1 / 3, -1 / 3, 1 / 4, -1 / 4, 3 / 4, -3 / 4, 2 / 3, -2 / 3, 3 / 2, -3 / 2, 5 / 2, -5 / 2, 5, -5}) → n_{pw}
  26. RandomDiscrete({1, 2, 3}, {2, 1, 1}) → pwright
  27. Random(-6, 6) → b_{pw}
  28. RandomDiscrete({b_{pw}, Random[-6, 6]}, {2, 1}) → b_{pw2}
  29. Random(-6, 6) → a_{pw}
  30. Expand((n_{pw} * (x - (a_{pw} + 1))^(2)) + b_{pw2} - n_{pw}) → f_{pwR3}
  31. Expand((n_{pw} * (x - a_{pw})) + b_{pw2}) → f_{pwR1}
  32. Expand((n_{pw} * (x - a_{pw})^(2)) + b_{pw2}) → f_{pwR2}
  33. RandomElement({-4, -3, -2, -1, 1, 2, 3, 4, 1 / 2, -1 / 2, 1 / 3, -1 / 3, 1 / 4, -1 / 4, 3 / 4, -3 / 4, 2 / 3, -2 / 3, 3 / 2, -3 / 2, 5 / 2, -5 / 2, 5, -5}) → m_{pw}
  34. Expand((m_{pw} * (x - (a_{pw} - 1))^(2)) + b_{pw} - m_{pw}) → f_{pwL3}
  35. Expand((m_{pw} * (x - a_{pw})^(2)) + b_{pw}) → f_{pwL2}
  36. RandomDiscrete({1, 2, 3}, {3, 2, 2}) → pwleft
  37. Expand((m_{pw} * (x - a_{pw})) + b_{pw}) → f_{pwL1}
  38. RandomDiscrete({-5, -3, -2, -1, 1, 2, 3, 5}, {1, 2, 3, 3, 3, 2, 2, 1}) → n
  39. Random(1, 20) → t_{1elem}
  40. If(t_{1elem} < 13.5, a_{trig}, a_{trignum} / a_{trigden}) → a_{advtrig}
  41. Element({sin(x), cos(x), tan(x), cot(x), sec(x), csc(x), sin((a * x)), 1 - cos(x), sin(x) - cos(x), sin(x)^(2), cos(x)^(2), sin((b * x)), 1 - tan(x), (m * x) + sin((π * x)), (n * x) + m + sin(((a * π) * x)), (b * x), x^(2), (a * x) + m, sin(((b * π) * x)), cos(((m * π) * x))}, t_{1elem}) → T_{n}
  42. Random(14, 20) → t_{2late}
  43. Random(1, 13) → t_{2early}
  44. If(t_{1elem} < 13.5, t_{2early}, t_{2late}) → t_{2elema}
  45. If(t_{1elem} ≠ t_{2elema}, t_{2elema}, t_{1elem} ≟ t_{2elema} ∧ t_{1elem} < 12.5, t_{1elem} + 1, t_{2elema} ≟ t_{1elem} ∧ t_{1elem} ≟ 13, 1, t_{2elema} ≟ t_{1elem} ∧ 13.5 < t_{1elem} < 19.5, t_{1elem} + 1, t_{1elem} ≟ t_{2elema} ∧ t_{1elem} ≟ 20, 14) → t_{2elem}
  46. Element({sin(x), cos(x), tan(x), cot(x), sec(x), csc(x), sin((a * x)), 1 - cos(x), sin(x) - cos(x), sin(x)^(2), cos(x)^(2), sin((b * x)), 1 - tan(x), (m * x) + sin((π * x)), (n * x) + m + sin(((a * π) * x)), (b * x), x^(2), (a * x) + m, sin(((b * π) * x)), cos(((m * π) * x))}, t_{2elem}) → T_{d1}
  47. RandomDiscrete({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1}) → spawwg
  48. Random(-5, 5) → a_{alg1}
  49. If(spawwg ≟ 5, Random[2, 5], spawwg ≟ 6, Random[2, 5], spawwg ≟ 11, Random[2, 5], spawwg ≟ 12, Random[2, 5], a_{alg1}) → a_{alg}
  50. RandomDiscrete({1, 2, 3, 4, 5, 6}, {4, 1, 1, 2, 1, 1}) → avwhatweget
  51. If(m ≟ 1, x - c, m ≟ -1, (-x) - c, (m * x) - c) → f_{AVden2}
  52. If(m ≟ 1, a - x, m ≟ -1, a + x, a - (m * x)) → f_{AVden3}
  53. If(m ≟ 1, x - a, m ≟ -1, (-x) - a, (m * x) - a) → f_{AVden}
  54. If(m ≟ 1, abs(x - a), m ≟ -1, abs((-x) - a), abs((m * x) - a)) → f_{AVnum}
  55. If(avwhatweget ≟ 1, f_{AVden}, avwhatweget ≟ 6, f_{AVnum}, avwhatweget ≟ 2, f_{AVden2}, avwhatweget ≟ 3, f_{AVden3}, avwhatweget ≟ 4, f_{AVnum}, avwhatweget ≟ 5, f_{AVnum}) → f_{AVd}
  56. If(avwhatweget ≟ 1, f_{AVnum}, avwhatweget ≟ 6, f_{AVden3}, avwhatweget ≟ 2, f_{AVnum}, avwhatweget ≟ 3, f_{AVnum}, avwhatweget ≟ 4, f_{AVden}, avwhatweget ≟ 5, f_{AVden2}) → f_{AVn}
  57. If(avwhatweget ≟ 1, LimitAbove[f_{AV}, a / m + 0.0001], avwhatweget ≟ 6, LimitAbove[f_{AV}, a / m + 0.0001], avwhatweget ≟ 2, LimitAbove[f_{AV}, a / m], avwhatweget ≟ 3, LimitAbove[f_{AV}, a / m + 0.0001], avwhatweget ≟ 4, LimitAbove[f_{AV}, a / m + 0.0001], avwhatweget ≟ 5, LimitAbove[f_{AV}, a / m]) → LimAVRight
  58. If(avwhatweget ≟ 1, LimitBelow[f_{AV}, a / m - 0.0001], avwhatweget ≟ 6, LimitBelow[f_{AV}, a / m - 0.0001], avwhatweget ≟ 2, LimitBelow[f_{AV}, a / m], avwhatweget ≟ 3, LimitBelow[f_{AV}, a / m - 0.0001], avwhatweget ≟ 4, LimitBelow[f_{AV}, a / m - 0.0001], avwhatweget ≟ 5, LimitBelow[f_{AV}, a / m]) → LimAVLeft
  59. RandomDiscrete({-10, -5, -3, -2, -1, 0, 1, 2, 3, 5, 10}, {1, 1, 3, 3, 1, 25, 1, 3, 3, 1, 1}) → g_{n}
  60. RandomDiscrete({-4, -3, -2, -1, 0, 1, 2, 3, 4}, {1, 2, 3, 4, 10, 4, 3, 2, 1}) → d_{n}
  61. DivisorsList(abs(c)) → c_{factor}
  62. Factors(abs(c)) →
  63. RandomElement(c_{factor}) → c_{facts}
  64. If(c ≟ 0, 1, c_{facts}) → c_{fact}
  65. DivisorsList(abs(a)) → a_{factors}
  66. Factors(abs(a)) →
  67. RandomElement(a_{factors}) → a_{fact}
  68. DivisorsList(abs(b)) → b_{factors}
  69. Factors(abs(b)) →
  70. RandomElement(b_{factors}) → b_{fact}
  71. Expand((m * (x - a))) → A_{d}
  72. Expand((m * (x - c))) → B_{n}
  73. Expand((n * (x - a))) → B_{d}
  74. Expand(((m * (x - a)) * (x - c))) → C_{n}
  75. Expand(((n * (x - a)) * (x - b))) → C_{d}
  76. Expand(((n * (x - a)) * (x - c))) → D_{n}
  77. Expand((m * (x - a))) → D_{d}
  78. Expand(((m * (x - a)) * (x - b))) → E_{d}
  79. Expand((abs(m) * (x - a))) → G_{n}
  80. Expand((abs(n) * (a - x))) → G_{d}
  81. RandomDiscrete({1, 2, 3, 4, 5, 6}, {1, 2, 2, 2, 1, 1}) → whatweget
  82. Limit(If[whatweget ≟ 1, A, whatweget ≟ 2, B, whatweget ≟ 3, C, whatweget ≟ 4, D, whatweget ≟ 5, E, whatweget ≟ 6, G], a) → RatExLimit
  83. If(whatweget ≟ 1, A_{n}, whatweget ≟ 6, G_{n}, whatweget ≟ 2, B_{n}, whatweget ≟ 3, C_{n}, whatweget ≟ 4, D_{n}, whatweget ≟ 5, E_{n}) → RatNum
  84. If(whatweget ≟ 1, A_{d}, whatweget ≟ 6, G_{d}, whatweget ≟ 2, B_{d}, whatweget ≟ 3, C_{d}, whatweget ≟ 4, D_{d}, whatweget ≟ 5, E_{d}) → RatDen
  85. Textfield(RatGuess) → InputBox1
  86. Textfield(RatGuessLeft) → InputBox3
  87. Textfield(RatGuessRight) → InputBox2
  88. Textfield(RatGuessValue) → InputBox4
  89. LimitBelow(If[whatweget ≟ 1, A, whatweget ≟ 2, B, whatweget ≟ 3, C, whatweget ≟ 4, D, whatweget ≟ 5, E, whatweget ≟ 6, G], a) → RatExLeft
  90. LimitAbove(If[whatweget ≟ 1, A, whatweget ≟ 2, B, whatweget ≟ 3, C, whatweget ≟ 4, D, whatweget ≟ 5, E, whatweget ≟ 6, G], a) → RatExRight
  91. FractionText(RatExLeft) → FracRatLeft
  92. FractionText(RatExRight) → FracRatRight
  93. FractionText(RatExLimit) → FracRatLim
  94. RandomPolynomial(Random[1, 6], -9, 9) → F_{d}
  95. RandomPolynomial(Random[1, 6], -9, 9) → F_{n}
  96. RandomDiscrete({-4, -3, -2, -1, 0, 1, 2, 3, 4}, {1, 2, 3, 4, 10, 4, 3, 2, 1}) → d_{d}
  97. Factor(((((m * x) - a) * ((d_{n} * x) + b / b_{fact})) * ((g_{n} * x) - c / c_{fact}))) → H_{n}
  98. RandomDiscrete({-10, -5, -3, -2, -1, 0, 1, 2, 3, 5, 10}, {1, 1, 3, 3, 1, 25, 1, 3, 3, 1, 1}) → g_{d}
  99. Factor(((((n * x) - b) * ((d_{d} * x) - a / a_{fact})) * ((g_{d} * x) + (2 * c)))) → H_{d}
  100. RandomDiscrete({1, 2, 3, 4, 5, 6, 7, 8}, {1, 1, 1, 1, 1, 5, 1, 7}) → infwhatweget
  101. Limit(If[infwhatweget ≟ 1, A, infwhatweget ≟ 2, B, infwhatweget ≟ 3, C, infwhatweget ≟ 4, D, infwhatweget ≟ 5, E, infwhatweget ≟ 6, F, infwhatweget ≟ 7, G, infwhatweget ≟ 8, H], Infinity) → RatInf
  102. Limit(If[infwhatweget ≟ 1, A, infwhatweget ≟ 2, B, infwhatweget ≟ 3, C, infwhatweget ≟ 4, D, infwhatweget ≟ 5, E, infwhatweget ≟ 6, F, infwhatweget ≟ 7, G, infwhatweget ≟ 8, H], -Infinity) → RatNegInf
  103. If(infwhatweget ≟ 1, A_{n}, infwhatweget ≟ 6, F_{n}, infwhatweget ≟ 7, G_{n}, infwhatweget ≟ 8, H_{n}, infwhatweget ≟ 2, B_{n}, infwhatweget ≟ 3, C_{n}, infwhatweget ≟ 4, D_{n}, infwhatweget ≟ 5, E_{n}) → RatInfNum
  104. If(infwhatweget ≟ 1, A_{d}, infwhatweget ≟ 6, F_{d}, infwhatweget ≟ 7, G_{d}, infwhatweget ≟ 8, H_{d}, infwhatweget ≟ 2, B_{d}, infwhatweget ≟ 3, C_{d}, infwhatweget ≟ 4, D_{d}, infwhatweget ≟ 5, E_{d}) → RatInfDen
  105. Textfield(RatNInfGuess) → InputBox5
  106. Textfield(RatPInfGuess) → InputBox6
  107. RandomDiscrete({1, 2, 3, 4, 5, 6}, {6, 5, 4, 3, 1, 1}) → trigwhatweget
  108. If(trigwhatweget ≟ 1, f_{trig1}, trigwhatweget ≟ 6, f_{trig6r}, trigwhatweget ≟ 2, f_{trig2}, trigwhatweget ≟ 3, f_{trig1r}, trigwhatweget ≟ 4, f_{trig2r}, trigwhatweget ≟ 5, f_{trig6}) → f_{trig0}
  109. If(trigwhatweget ≟ 1, f_{trig3}, trigwhatweget ≟ 6, f_{trig4r}, trigwhatweget ≟ 2, f_{trig5}, trigwhatweget ≟ 3, f_{trig4}, trigwhatweget ≟ 4, f_{trig3r}, trigwhatweget ≟ 5, f_{trig5r}) → f_{trigv}
  110. Limit(f_{trig0}, 0) → LimTrig0
  111. Limit(f_{trigv}, a_{trig}) → LimTrigV
  112. RandomDiscrete({1, 2, 3}, {5, 3, 1}) → trigwhatweget2
  113. If(trigwhatweget ≟ 1, f_{trig1}, trigwhatweget ≟ 6, f_{trig1r}, trigwhatweget ≟ 2, f_{trig1r}, trigwhatweget ≟ 3, f_{trig1}, trigwhatweget ≟ 4, f_{trig1r}, trigwhatweget ≟ 5, f_{trig1}) → f_{trigsneak}
  114. Limit(f_{trigsneak}, b_{trig}) → LimTrigSneak
  115. If(trigwhatweget2 ≟ 1, LimTrig0, trigwhatweget2 ≟ 2, LimTrigV, trigwhatweget2 ≟ 3, round(LimTrigSneak, 3)) → triggoesto
  116. If(trigwhatweget2 ≟ 1, f_{trig0}, trigwhatweget2 ≟ 2, f_{trigv}, trigwhatweget2 ≟ 3, f_{trigsneak}) → f_{trig}
  117. Textfield(LimTrigGuess) → InputBox7
  118. Textfield(AVLeftGuess) → InputBox8
  119. Textfield(AVRightGuess) → InputBox9
  120. Textfield(AVLimitGuess) → InputBox10
  121. If(LimAVLeft ≟ LimAVRight, LimAVLeft, NaN) → LimAV
  122. FractionText(LimAVLeft) → avleftanswer
  123. FractionText(LimAVRight) → avrightanswer
  124. FractionText(LimAV) → avlimanswer
  125. RandomElement({(a * x^(2)) + (b * x) + c, (a * x) + b, ((a * (x - c)) * (x - b)), (x - b)^(2), (((m * x) - b) * (x - c))}) → f_{dirn}
  126. RandomElement({(b * x^(2)) + (c * x) + a, (m * x) + c, (m * (x - c)^(2)), ((x - c) * (x - m)), (((a * x) - m) * ((c * x) - b))}) → f_{dird}
  127. Limit(f_{dir}, a) → DirLim
  128. FractionText(DirLim) → directlimanswer
  129. Textfield(DirectGuess) → InputBox11
  130. If(-1000 < FactorLimitA < 1000, FactorLimitA, FactorLimitA ≟ NaN, 999, 999) → FactorLimit
  131. Textfield(FactorGuess) → InputBox12
  132. Random(a_{alg}^(2) + 1, 29) → c_{alg}
  133. If(a_{alg} ≟ 1, 0, a_{alg} ≟ -1, 0, Random[1, a_{alg}^(2) - 1]) → d_{alg}
  134. Random(a_{alg} + 1, 15) → b_{alg}
  135. If(spawwg ≟ 1, sqrt(x + b_{alg}) - Simplify[sqrt(b_{a})], spawwg ≟ 6, x - a_{alg}, spawwg ≟ 7, 1 / (x + b_{alg}) - 1 / b_{a}, spawwg ≟ 8, x - a_{alg}, spawwg ≟ 9, 1 / (c_{alg} - x^(2)) - 1 / c_{a}, spawwg ≟ 10, x - a_{alg}, spawwg ≟ 2, spawwg ≟ 11, 1 / (x^(2) - d_{alg}) - 1 / d_{a}, spawwg ≟ 12, x - a_{alg}, x - a_{alg}, spawwg ≟ 3, sqrt(c_{alg} - x^(2)) - Simplify[sqrt(c_{a})], spawwg ≟ 4, x - a_{alg}, spawwg ≟ 5, sqrt(x^(2) - d_{alg}) - Simplify[sqrt(d_{a})]) → f_{algN}
  136. If(spawwg ≟ 1, x - a_{alg}, spawwg ≟ 6, sqrt(x^(2) - d_{alg}) - Simplify[sqrt(d_{a})], spawwg ≟ 7, x - a_{alg}, spawwg ≟ 8, 1 / (x + b_{alg}) - 1 / b_{a}, spawwg ≟ 9, x - a_{alg}, spawwg ≟ 10, 1 / (c_{alg} - x^(2)) - 1 / c_{a}, spawwg ≟ 2, spawwg ≟ 11, x - a_{alg}, spawwg ≟ 12, 1 / (x^(2) - d_{alg}) - 1 / d_{a}, sqrt(x + b_{alg}) - Simplify[sqrt(b_{a})], spawwg ≟ 3, x - a_{alg}, spawwg ≟ 4, sqrt(c_{alg} - x^(2)) - Simplify[sqrt(c_{a})], spawwg ≟ 5, x - a_{alg}) → f_{algD}
  137. Limit(f_{alg}, a_{alg}) → AlgLim
  138. Textfield(AlgGuess) → InputBox13
  139. If(trigwhatweget2 ≟ 1, LimTrig0, trigwhatweget2 ≟ 2, LimTrigV, trigwhatweget2 ≟ 3, LimTrigSneak) → LimTrig_{a}
  140. FractionText(triggoesto) → fractiontriggoesto
  141. Text(triggoesto) → triggoesto3
  142. If(-50 < LimAV < 50, 1, 0) → LimAVund
  143. If(-1000 < LimTrig_{a} < 1000, LimTrig_{a}, 999) → LimTrig
  144. SurdText(FactorLimit) → factoranswer
  145. SurdText(AlgLim) → alganswer
  146. If(-1000 < Limit[T, a_{advtrig}] < 1000, Limit[T, a_{advtrig}], 999) → AdvTrigLimit
  147. SurdText(AdvTrigLimit) → advtriganswer
  148. Textfield(AdvTrigGuess) → InputBox14
  149. If(pwright ≟ 1, f_{pwR1}, pwright ≟ 2, f_{pwR2}, f_{pwR3}) → f_{pwR}
  150. If(pwleft ≟ 1, f_{pwL1}, pwleft ≟ 2, f_{pwL2}, f_{pwL3}) → f_{pwL}
  151. RandomDiscrete({b_{pw2}, Random[-6, 6]}, {2, 1}) → b_{pw3}
  152. RandomElement({f_{pw2part}, f_{pw2part2}, f_{pw3part}}) → f_{pw}
  153. LimitBelow(f_{pwL}, a_{pw}) → PWLimLeft
  154. LimitAbove(f_{pwR}, a_{pw}) → PWLimRight
  155. If(PWLimLeft ≟ PWLimRight, PWLimLeft, 999) → PWLim
  156. Textfield(PWContGuess) → InputBox15
  157. Text(PWLimLeft) → pwleftanswer
  158. Text(PWLimRight) → pwrightanswer
  159. Text(PWLim) → pwlimanswer
  160. Textfield(PWLeftGuess) → InputBox16
  161. Textfield(PWRightGuess) → InputBox17
  162. Textfield(PWLimitGuess) → InputBox18
  163. If(PWLim ≟ f_{pw}(a_{pw}), "yes", "no") → pwcontanswer
  164. Textfield(PWValueGuess) → InputBox19
  165. Text(f_{pw}(a_{pw})) → pwvaluetext
  166. TableText({"x", a_{pw} - 0.1, a_{pw} - 0.01, a_{pw} - 0.001, a_{pw}, a_{pw} + 0.001, a_{pw} + 0.01, a_{pw} + 0.1}, {"f(x)", f_{pw}(a_{pw} - 0.01), f_{pw}(a_{pw} - 0.01), f_{pw}(a_{pw} - 0.001), f_{pw}(a_{pw}), f_{pw}(a_{pw} + 0.001), f_{pw}(a_{pw} + 0.01), f_{pw}(a_{pw} + 0.1)}, "hc|_") → texttable
  167. If(derwhatweget ≟ 1, sin(x - c_{trig}), derwhatweget ≟ 6, cot(x - c_{trig}), derwhatweget ≟ 7, ln(x + a_{pos}), derwhatweget ≟ 12, (m * ℯ^((b_{2} * x) - a_{1})), derwhatweget ≟ 9, ((m * x) - a_{1})^(2), derwhatweget ≟ 10, (m * (x - a_{1})^(2)), derwhatweget ≟ 2, derwhatweget ≟ 11, ((m * x) - a_{1})^(3), derwhatweget ≟ 8, sqrt(x + a_{pos}), derwhatweget ≟ 13, m / (x - a_{1}), derwhatweget ≟ 14, m / (x - a_{1})^(2), derwhatweget ≟ 15, (m * x^(2)) + (b_{1} * x), cos(x - c_{trig}), derwhatweget ≟ 16, log(logbase, x + a_{pos}), derwhatweget ≟ 3, csc(x - c_{trig}), derwhatweget ≟ 4, sec(x - c_{trig}), derwhatweget ≟ 5, tan(x - c_{trig})) → f
  168. If(-100000 ≤ f'(a_{der}) ≤ 100000, f'(a_{der}), f'(a_{der}) ≟ NaN, 999, 999) → DerValue
  169. SurdText(DerValue) → dervaluetext
  170. Random(1, 2) → dertype
  171. Textfield(DerGuess) → InputBox20
  172. If(1 ≤ derwhatweget ≤ 6, a_{der} - c_{trig}, derwhatweget ≟ 12, (b_{2} * a_{der}) - a_{1}, derwhatweget ≟ 13, a_{der} - a_{1}, derwhatweget ≟ 14, a_{der} - a_{1}, derwhatweget ≟ 15, a_{der}, derwhatweget ≟ 16, a_{der} + a_{pos}, 7 ≤ derwhatweget ≤ 8, a_{der} + a_{pos}, derwhatweget ≟ 9, (m * a_{der}) - a_{1}, derwhatweget ≟ 10, a_{der} - a_{1}, derwhatweget ≟ 11, (m * a_{der}) - a_{1}) → InsideNumber
  173. If(d ≟ 1, sin(InsideNumber + h), d ≟ 6, cot(InsideNumber + h), d ≟ 7, ln(InsideNumber + h), d ≟ 8, sqrt(InsideNumber + h), d ≟ 9, (InsideNumber + h)^(2), d ≟ 12, (m * ℯ^(InsideNumber + h)), d ≟ 2, d ≟ 11, (InsideNumber + h)^(3), d ≟ 10, (m * (InsideNumber + h)^(2)), d ≟ 13, m / (InsideNumber + h), d ≟ 14, m / (InsideNumber + h)^(2), d ≟ 15, (m * (InsideNumber + h)^(2)) + (b_{1} * (InsideNumber + h)), cos(InsideNumber + h), d ≟ 16, log(logbase, InsideNumber + h), d ≟ 3, csc(InsideNumber + h), d ≟ 4, sec(InsideNumber + h), d ≟ 5, tan(InsideNumber + h)) → fofaplush
  174. If(d ≟ 1, Simplify[sin(InsideNumber)], d ≟ 6, Simplify[cot(InsideNumber)], d ≟ 7, Simplify[ln(InsideNumber)], d ≟ 8, Simplify[sqrt(InsideNumber)], d ≟ 9, InsideNumber^(2), d ≟ 10, (m * InsideNumber^(2)), d ≟ 2, d ≟ 11, InsideNumber^(3), d ≟ 12, (m * Simplify[ℯ^(InsideNumber)]), d ≟ 13, Simplify[m / InsideNumber], d ≟ 14, Simplify[m / InsideNumber^(2)], d ≟ 15, (m * InsideNumber^(2)) + (b_{1} * InsideNumber), Simplify[cos(InsideNumber)], d ≟ 16, Simplify[log(logbase, InsideNumber)], d ≟ 3, Simplify[csc(InsideNumber)], d ≟ 4, Simplify[sec(InsideNumber)], d ≟ 5, Simplify[tan(InsideNumber)]) → g
  175. Random(1, 11) → fundomwhatweget
  176. Random(1, 12) → fundomwhatweget2a
  177. If(fundomwhatweget ≟ 1, RandomPolynomial[3, -4, 6], fundomwhatweget ≟ 6, (Random[2, 8] * sqrt(x - a_{1})), fundomwhatweget ≟ 7, (Random[2, 8] * cbrt(x - b_{1})), fundomwhatweget ≟ 8, log(logbase, x + a_{pos}), fundomwhatweget ≟ 9, RandomPolynomial[4, -7, 7], fundomwhatweget ≟ 10, x^(1 / Random[2, 5]), fundomwhatweget ≟ 2, fundomwhatweget ≟ 11, logbase^(x) + RandomPolynomial[2, -4, 4], fundomwhatweget ≟ 12, ℯ^((Random[2, 8] * x) + m), (Random[2, 8] * ℯ^(Simplify[(m * x) - a_{1}])), fundomwhatweget ≟ 3, (Random[2, 8] * ℯ^(x)) + RandomPolynomial[2, -5, 8], fundomwhatweget ≟ 4, (Random[2, 9] * ln(x + a_{pos})), fundomwhatweget ≟ 5, Random[2, 5]^(x)) → fundomnum
  178. If(fundomwhatweget2a ≠ fundomwhatweget, fundomwhatweget2a, 12) → fundomwhatweget2
  179. If(fundomwhatweget2 ≟ 1, RandomPolynomial[3, -4, 6], fundomwhatweget2 ≟ 6, (Random[2, 8] * sqrt(x - a_{1})), fundomwhatweget2 ≟ 7, (Random[2, 8] * cbrt(x - b_{1})), fundomwhatweget2 ≟ 8, log(logbase, x + a_{pos}), fundomwhatweget2 ≟ 9, RandomPolynomial[4, -7, 7], fundomwhatweget2 ≟ 10, x^(1 / Random[2, 5]), fundomwhatweget2 ≟ 2, fundomwhatweget2 ≟ 11, logbase^(x) + RandomPolynomial[2, -4, 4], fundomwhatweget2 ≟ 12, ℯ^((Random[2, 8] * x) + m), (Random[2, 8] * ℯ^(Simplify[(m * x) - a_{1}])), fundomwhatweget2 ≟ 3, (Random[2, 8] * ℯ^(x)) + RandomPolynomial[2, -5, 8], fundomwhatweget2 ≟ 4, (Random[2, 9] * ln(x + a_{pos})), fundomwhatweget2 ≟ 5, Random[2, 5]^(x)) → fundomden
  180. If(-100000 < Limit[FunDom, Infinity] < 100000, Limit[FunDom, Infinity], Limit[FunDom, Infinity] ≟ Infinity, Infinity, Limit[FunDom, Infinity] ≟ -Infinity, -Infinity, 999) → FunDomLim
  181. FractionText(FunDomLim) → fundomanswer
  182. Textfield(FunDomGuess) → InputBox21
  183. RandomDiscrete({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, {2, 1, 2, 2, 1, 3, 1, 1, 3, 3, 3, 2, 3}) → RandomButton