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tan(x)/(x^2*cot(3*x))

Limit of the function tan(x)/(x^2*cot(3*x))

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     /   tan(x)  \
 lim |-----------|
x->0+| 2         |
     \x *cot(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
Limit(tan(x)/((x^2*cot(3*x))), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(3 x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(\frac{x^{2}}{\tan{\left(x \right)}}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(3 x \right)}}}{\frac{d}{d x} \frac{x^{2}}{\tan{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cot^{2}{\left(3 x \right)} + 3}{\left(- x^{2} - \frac{x^{2}}{\tan^{2}{\left(x \right)}} + \frac{2 x}{\tan{\left(x \right)}}\right) \cot^{2}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cot^{2}{\left(3 x \right)} + 3}{\left(- x^{2} - \frac{x^{2}}{\tan^{2}{\left(x \right)}} + \frac{2 x}{\tan{\left(x \right)}}\right) \cot^{2}{\left(3 x \right)}}\right)$$
=
$$3$$
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
Rapid solution [src]
3
$$3$$
One‐sided limits [src]
     /   tan(x)  \
 lim |-----------|
x->0+| 2         |
     \x *cot(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
3
$$3$$
= 3.0
     /   tan(x)  \
 lim |-----------|
x->0-| 2         |
     \x *cot(3*x)/
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
3
$$3$$
= 3.0
= 3.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right) = 3$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right) = \tan{\left(1 \right)} \tan{\left(3 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right) = \tan{\left(1 \right)} \tan{\left(3 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
More at x→-oo
Numerical answer [src]
3.0
3.0
The graph
Limit of the function tan(x)/(x^2*cot(3*x))
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