Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (3-x)^(x/(2-x))
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Limit of 3+x+x^(3/2)-x^2
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Limit of 3*x*cot(9*x)
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Limit of -3+x+3*x^3-x^2/2
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Identical expressions
- three *x*cot(nine *x)
- 3 multiply by x multiply by cotangent of (9 multiply by x)
- three multiply by x multiply by cotangent of (nine multiply by x)
- 3xcot(9x)
- 3xcot9x
Limit of the function 3*x*cot(9*x)
The solution
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[src]
lim (3*x*cot(9*x)) x->0+
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right)$$
Limit((3*x)*cot(9*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(3 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(9 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 3 x}{\frac{d}{d x} \frac{1}{\cot{\left(9 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cot^{2}{\left(9 x \right)}}{9 \cot^{2}{\left(9 x \right)} + 9}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cot^{2}{\left(9 x \right)}}{9 \cot^{2}{\left(9 x \right)} + 9}\right)$$
=
$$\frac{1}{3}$$
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
$$\lim_{x \to 0^+}\left(3 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(9 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 3 x}{\frac{d}{d x} \frac{1}{\cot{\left(9 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cot^{2}{\left(9 x \right)}}{9 \cot^{2}{\left(9 x \right)} + 9}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cot^{2}{\left(9 x \right)}}{9 \cot^{2}{\left(9 x \right)} + 9}\right)$$
=
$$\frac{1}{3}$$
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
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lim (3*x*cot(9*x)) x->0+
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
lim (3*x*cot(9*x)) x->0-
$$\lim_{x \to 0^-}\left(3 x \cot{\left(9 x \right)}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
= 0.333333333333333
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(3 x \cot{\left(9 x \right)}\right) = \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(3 x \cot{\left(9 x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(3 x \cot{\left(9 x \right)}\right) = \frac{3}{\tan{\left(9 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(3 x \cot{\left(9 x \right)}\right) = \frac{3}{\tan{\left(9 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(3 x \cot{\left(9 x \right)}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(3 x \cot{\left(9 x \right)}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(3 x \cot{\left(9 x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(3 x \cot{\left(9 x \right)}\right) = \frac{3}{\tan{\left(9 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(3 x \cot{\left(9 x \right)}\right) = \frac{3}{\tan{\left(9 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(3 x \cot{\left(9 x \right)}\right)$$
More at x→-oo
The graph