Mister Exam
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How to use it?
- Limit of the function:
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Limit of (1+x^2+9*x)/(-5+2*x+7*x^2)
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Limit of (-tan(2*x)+sin(2*x))/x^3
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Limit of 3/n^4
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Limit of (1-cos(x)^2)/(x^2*sin(x)^2)
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Identical expressions
- sin(tan(two *x))/asin(three *x)
- sinus of ( tangent of (2 multiply by x)) divide by arc sinus of e of (3 multiply by x)
- sinus of ( tangent of (two multiply by x)) divide by arc sinus of e of (three multiply by x)
- sin(tan(2x))/asin(3x)
- sintan2x/asin3x
- sin(tan(2*x)) divide by asin(3*x)
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Similar expressions
- sin(tan(2*x))/arcsin(3*x)
Limit of the function sin(tan(2*x))/asin(3*x)
The solution
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[src]
/sin(tan(2*x))\ lim |-------------| x->0+\ asin(3*x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
Limit(sin(tan(2*x))/asin(3*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(\tan{\left(2 x \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \operatorname{asin}{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\tan{\left(2 x \right)} \right)}}{\frac{d}{d x} \operatorname{asin}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - 9 x^{2}} \left(2 \tan^{2}{\left(2 x \right)} + 2\right) \cos{\left(\tan{\left(2 x \right)} \right)}}{3}\right)$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\frac{2}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(\tan{\left(2 x \right)} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \operatorname{asin}{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\tan{\left(2 x \right)} \right)}}{\frac{d}{d x} \operatorname{asin}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - 9 x^{2}} \left(2 \tan^{2}{\left(2 x \right)} + 2\right) \cos{\left(\tan{\left(2 x \right)} \right)}}{3}\right)$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\frac{2}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
One‐sided limits
[src]
/sin(tan(2*x))\ lim |-------------| x->0+\ asin(3*x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
/sin(tan(2*x))\ lim |-------------| x->0-\ asin(3*x) /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
= 0.666666666666667
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{2}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{\sin{\left(\tan{\left(2 \right)} \right)}}{\operatorname{asin}{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{\sin{\left(\tan{\left(2 \right)} \right)}}{\operatorname{asin}{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{\sin{\left(\tan{\left(2 \right)} \right)}}{\operatorname{asin}{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right) = \frac{\sin{\left(\tan{\left(2 \right)} \right)}}{\operatorname{asin}{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\tan{\left(2 x \right)} \right)}}{\operatorname{asin}{\left(3 x \right)}}\right)$$
More at x→-oo
The graph