Übung: Grenzwerte #1
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\large\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: "#00A6E2"
}, function() {
plot( function(x) {
return q_lcoef * x * x;
}, [-10, 10] );
circle( [a, limtoa], 4 / 20, { fill: "white" } )
});
limtoa
0randRangeNonZero(-3, 3)randRangeNonZero(-3, 3)q_lcoefl_conscurFunc(0)- Existiert nicht.
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 \large \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: "#00A6E2"
}, 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" } )
});
Existiert nicht.
0aabs_consabs_coefabs_coef * -1abs_coef * abs_coefabs_coef * abs_coef * -1randRangeNonZero(-3, 3)randRangeNonZero(-3, 3)
Der linksseitige Grenzwert entspricht nicht dem rechtsseitigen Grenzwert. Daher hat f(x) keinen Grenzwert für 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
$._("wenn")
$._("sonst")
Für \displaystyle\large f(x) = \begin{cases}
d_cons & \text{IF } x = a \\
d_line & \text{OTHERWISE}
\end{cases}, bestimme \displaystyle\large \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: "#00A6E2"
}, 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: "#00A6E2", stroke: "none" } )
});
limtoa
0ad_consl_coeffractionReduce(l_cons * -1, l_coef)randRangeNonZero(-3, 3)randRangeNonZero(-3, 3)- Existiert nicht.
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\large \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: "#00A6E2"
}, function() {
plot( function(x) {
return q_lcoef * x * x + q_cons;
}, [-10, 10] );
});
limtoa
0curFunc(0)randRangeNonZero(-3, 3)q_consq_cons * -1- Existiert nicht.
Was passiert, wenn wir uns x = a von links annähern?
line( [a - 2, 0], [a, 0], {
stroke: "#EF2978",
arrows: "->"
});
x | localeToFixed(a - 0.1, -1) | localeToFixed(a - 0.01, -1) | localeToFixed(a - 0.001, -1) |
|---|---|---|---|
f(x) | localeToFixed(curFunc(a - 0.1), 4) | localeToFixed(curFunc(a - 0.01), 4) | localeToFixed(curFunc(a - 0.001), 4) |
f(x) a von links aus annähert, desto mehr nähert es sich l_limtoa.
Wenn wir uns von rechts aus x = a annähern, bekommen wir:
x | localeToFixed(a + 0.1, -1) | localeToFixed(a + 0.01, -1) | localeToFixed(a + 0.001, -1) |
|---|---|---|---|
f(x) | localeToFixed(curFunc(a + 0.1), 4) | localeToFixed(curFunc(a + 0.01), 4) | localeToFixed(curFunc(a + 0.001), 4) |
Es sieht so aus, dass je weiter sich f(x) von rechts aus annähert, desto mehr nähert es sich r_limtoa.
line( [a + 2, 0], [a, 0], {
stroke: "#EF2978",
arrows: "->"
});
Daher ist der Grenzwert limtoa.