randRangeNonZero(-3, 3) floor(sqrt(abs(10 / q_lcoef))) randRangeNonZero(-1 * edge, edge) q_lcoef + 'x^2' cleanMath('(x + ' + l_cons + ')') function(x) { return truncate_to_max(q_lcoef * Math.pow(x, 2), 4); } -1 * l_cons curFunc(a) limtoa limtoa

\displaystyle\lim_{x\toa} \dfrac{quadraticLINE}{LINE} = {?}

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); style({ stroke: "#6495ed" }, function() { plot( function(x) { return q_lcoef * x * x; }, [-10, 10] ); circle( [a, limtoa], 4 / 20, { fill: "white" } ) });

limtoa

  • 0
  • randRangeNonZero(-3, 3)
  • randRangeNonZero(-3, 3)
  • q_lcoef
  • l_cons
  • curFunc(0)
  • Nie istnieje.
randRangeNonZero(-5, 5) randRangeNonZero(-7, 7) function(x) { return x + abs_cons > 0 ? abs_coef : abs_coef * -1; } abs_cons * -1 curFunc(a) abs_coef * -1 abs_coef

\displaystyle \lim_{x\toa} \dfrac{abs_coef|x + abs_cons|}{x + abs_cons} = {?}

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); style({ stroke: "#6495ed" }, function() { line( [-11, abs_coef * -1], [-abs_cons, abs_coef * -1] ); line( [-abs_cons, abs_coef], [11, abs_coef] ); circle( [-abs_cons, -abs_coef], 4 / 20, { fill: "white" } ) circle( [-abs_cons, abs_coef], 4 / 20, { fill: "white" } ) });

Nie istnieje.

  • 0
  • a
  • abs_cons
  • abs_coef
  • abs_coef * -1
  • abs_coef * abs_coef
  • abs_coef * abs_coef * -1
  • randRangeNonZero(-3, 3)
  • randRangeNonZero(-3, 3)

Granica z lewej strony nie jest równa granicy z prawej strony, więc f(x) nie ma granicy x \to a.

randRangeNonZero(-3, 3) randRangeNonZero(-4, 4) randRangeNonZero(-7, 7) expr(["+", ["*", l_coef, "x"], l_cons]) function(x) { return l_coef * x + l_cons; } ceil((-10 - l_cons)/l_coef) floor((10 - l_cons)/l_coef) l_coef > 0 ? randRangeNonZero(a0, a1) : randRangeNonZero(a1, a0) curFunc(a) limtoa limtoa

Jeżeli f(x) = \begin{cases} d_cons & \text{if $x = a$} \\ d_line & \text{otherwise} \end{cases}, oblicz \displaystyle \lim_{x \to a} f(x).

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); style({ stroke: "#6495ed" }, function() { plot( function(x) { return l_coef * x + l_cons; }, [-10, 10] ); circle( [a, limtoa], 4 / 20, { fill: "white" } ) circle( [a, d_cons], 4 / 20, { fill: "#6495ed", stroke: "none" } ) });

limtoa

  • 0
  • a
  • d_cons
  • l_coef
  • fractionReduce(l_cons * -1, l_coef)
  • randRangeNonZero(-3, 3)
  • randRangeNonZero(-3, 3)
  • Nie istnieje.
randRangeNonZero(-3, 3) randRangeNonZero(-3, 3) floor(sqrt(abs((10 - q_cons) / q_lcoef))) randRangeNonZero(-1 * edge, edge) function(x) { return q_lcoef * Math.pow(x, 2) + q_cons; } curFunc(a) limtoa limtoa

\displaystyle \lim_{x\toa} expr(["+", ["*", q_lcoef, ["^", "x", 2]], q_cons]) = {?}

graphInit({ range: 10, scale: 20, tickStep: 1, labelStep: 1, unityLabels: false, labelFormat: function( s ) { return "\\small{" + s + "}"; }, axisArrows: "<->" }); style({ stroke: "#6495ed" }, function() { plot( function(x) { return q_lcoef * x * x + q_cons; }, [-10, 10] ); });

limtoa

  • 0
  • curFunc(0)
  • randRangeNonZero(-3, 3)
  • q_cons
  • q_cons * -1
  • Nie istnieje.

Co się dzieje

line( [a - 2, 0], [a, 0], { stroke: "#ff00af", arrows: "->" });
xa - 0.1a - 0.01a - 0.001
f(x)curFunc(a - 0.1).toFixed(4)curFunc(a - 0.01).toFixed(4)curFunc(a - 0.001).toFixed(4)
It looks like f(x) is approaching l_limtoa from the left.

Kiedy zbliżamy się do x = a z prawej strony, otrzymujemy:

xa + 0.1a + 0.01a + 0.001
f(x)curFunc(a + 0.1).toFixed(4)curFunc(a + 0.01).toFixed(4)curFunc(a + 0.001).toFixed(4)
It looks like f(x) is approaching r_limtoa from the right.
line( [a + 2, 0], [a, 0], { stroke: "#ff00af", arrows: "->" });

Więc granica wynosi limtoa.