Mister Exam

### Other calculators:

• #### How to use it?

• Limit of the function:
• Limit of tan(x)/(x^2*cot(3*x))
• Limit of z*sin(1/z)
• Limit of tanh(x)
• Limit of log(log(x))
• #### Identical expressions

• tan(x)/(x^ two *cot(three *x))
• tangent of (x) divide by (x squared multiply by cotangent of (3 multiply by x))
• tangent of (x) divide by (x to the power of two multiply by cotangent of (three multiply by x))
• tan(x)/(x2*cot(3*x))
• tanx/x2*cot3*x
• tan(x)/(x²*cot(3*x))
• tan(x)/(x to the power of 2*cot(3*x))
• tan(x)/(x^2cot(3x))
• tan(x)/(x2cot(3x))
• tanx/x2cot3x
• tanx/x^2cot3x
• tan(x) divide by (x^2*cot(3*x))

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

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### The solution

You have entered [src]
     /   tan(x)  \
lim |-----------|
x->0+| 2         |
\x *cot(3*x)/
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
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
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x^{2} \cot{\left(3 x \right)}\right) = 0$$
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
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(x \right)}}{\frac{d}{d x} x^{2} \cot{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)} + 1}{x^{2} \left(- 3 \cot^{2}{\left(3 x \right)} - 3\right) + 2 x \cot{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)} + 1}{x^{2} \left(- 3 \cot^{2}{\left(3 x \right)} - 3\right) + 2 x \cot{\left(3 x \right)}}\right)$$
=
$$3$$
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
     /   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
= 3
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
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right) = \frac{\tan{\left(1 \right)}}{\cot{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(x \right)}}{x^{2} \cot{\left(3 x \right)}}\right) = \frac{\tan{\left(1 \right)}}{\cot{\left(3 \right)}}$$
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
3.0
3.0