Mister Exam
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How to use it?
- Limit of the function:
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Limit of 2^(-n)*2^(1+n)
- Limit of tan(k*x)/x
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Limit of (-x+tan(x))/(x+2*sin(x))
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Limit of asin(5*x)/tan(3*x)
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Identical expressions
- asin(five *x)/tan(three *x)
- arc sinus of e of (5 multiply by x) divide by tangent of (3 multiply by x)
- arc sinus of e of (five multiply by x) divide by tangent of (three multiply by x)
- asin(5x)/tan(3x)
- asin5x/tan3x
- asin(5*x) divide by tan(3*x)
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Similar expressions
- x*(-3+x)*asin(5*x)/(tan(3*x)*tan(2*pi*x))
- arcsin(5*x)/tan(3*x)
Limit of the function asin(5*x)/tan(3*x)
The solution
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[src]
/asin(5*x)\ lim |---------| x->0+\ tan(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
Limit(asin(5*x)/tan(3*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \operatorname{asin}{\left(5 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(5 x \right)}}{\frac{d}{d x} \tan{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{\sqrt{1 - 25 x^{2}} \left(3 \tan^{2}{\left(3 x \right)} + 3\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{3 \tan^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{3 \tan^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\frac{5}{3}$$
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
$$\lim_{x \to 0^+} \operatorname{asin}{\left(5 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(5 x \right)}}{\frac{d}{d x} \tan{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{\sqrt{1 - 25 x^{2}} \left(3 \tan^{2}{\left(3 x \right)} + 3\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{3 \tan^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{3 \tan^{2}{\left(3 x \right)} + 3}\right)$$
=
$$\frac{5}{3}$$
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]
/asin(5*x)\ lim |---------| x->0+\ tan(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
5/3
$$\frac{5}{3}$$
= 1.66666666666667
/asin(5*x)\ lim |---------| x->0-\ tan(3*x)/
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
5/3
$$\frac{5}{3}$$
= 1.66666666666667
= 1.66666666666667
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{5}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{5}{3}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{\tan{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{\tan{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{5}{3}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{\tan{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{\tan{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{\tan{\left(3 x \right)}}\right)$$
More at x→-oo
The graph