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- Limit of the function:
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-
Limit of (-x^2+4*x)/(2-sqrt(x))
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Limit of ((2+3*x)/(-1+3*x))^(-1+4*x)
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Limit of (3-x+2*x^2)/(5+x^3-8*x)
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Identical expressions
- cot(x)*sin(two *x)
- cotangent of (x) multiply by sinus of (2 multiply by x)
- cotangent of (x) multiply by sinus of (two multiply by x)
- cot(x)sin(2x)
- cotxsin2x
-
Similar expressions
- cot(x)^sin(2*x)
Limit of the function cot(x)*sin(2*x)
The solution
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lim (cot(x)*sin(2*x)) x->0+
$$\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
Limit(cot(x)*sin(2*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cos{\left(2 x \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$2$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cos{\left(2 x \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\left(2 \cot^{2}{\left(x \right)} + 2\right) \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(x \right)} + 2}}{\frac{d}{d x} \left(- \frac{\cot^{4}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \cot^{2}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{1}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(2 \cot^{2}{\left(x \right)} + 2\right)^{2} \left(- \frac{\left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{4}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{2 \left(4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}\right) \cot^{2}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{4 \left(- 6 \cot^{2}{\left(x \right)} - 6\right) \cot^{5}{\left(x \right)} + 4 \left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{\left(- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}\right)^{2}} - \frac{\left(- 4 \cot^{2}{\left(x \right)} - 4\right) \cot^{3}{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}} - \frac{2 \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{- 4 \cot^{6}{\left(x \right)} - 4 \cot^{4}{\left(x \right)}}\right)}\right)$$
=
$$2$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
One‐sided limits
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lim (cot(x)*sin(2*x)) x->0+
$$\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
2
$$2$$
= 2.0
lim (cot(x)*sin(2*x)) x->0-
$$\lim_{x \to 0^-}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
2
$$2$$
= 2.0
= 2.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = 2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right) = \frac{\sin{\left(2 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cot{\left(x \right)}\right)$$
More at x→-oo
The graph