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cot(x)*log(x)*sin(x)
Limit of the function/ cot(x)*log(x)*sin(x)

Limit of the function cot(x)*log(x)*sin(x)

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You have entered [src] 
 lim (cot(x)*log(x)*sin(x))
x->0+
lim⁡x→0+(log⁡(x)cot⁡(x)sin⁡(x))\lim_{x \to 0^+}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) 
Limit((cot(x)*log(x))*sin(x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+(log⁡(x)sin⁡(x))=0\lim_{x \to 0^+}\left(\log{\left(x \right)} \sin{\left(x \right)}\right) = 0
and limit for the denominator is
lim⁡x→0+1cot⁡(x)=0\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→0+(log⁡(x)cot⁡(x)sin⁡(x))\lim_{x \to 0^+}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(log⁡(x)sin⁡(x)cot⁡(x))\lim_{x \to 0^+}\left(\log{\left(x \right)} \sin{\left(x \right)} \cot{\left(x \right)}\right)
=
lim⁡x→0+(ddxlog⁡(x)sin⁡(x)ddx1cot⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)
=
lim⁡x→0+(log⁡(x)cos⁡(x)+sin⁡(x)x1+1cot⁡2(x))\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}}{1 + \frac{1}{\cot^{2}{\left(x \right)}}}\right)
=
lim⁡x→0+(log⁡(x)cos⁡(x)+sin⁡(x)x1+1cot⁡2(x))\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}}{1 + \frac{1}{\cot^{2}{\left(x \right)}}}\right)
=
−∞-\infty
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
02468-8-6-4-2-1010-1010
Plot the graph
Rapid solution [src] 
-oo
−∞-\infty
Expand and simplify
One‐sided limits [src] 
 lim (cot(x)*log(x)*sin(x))
x->0+
lim⁡x→0+(log⁡(x)cot⁡(x)sin⁡(x))\lim_{x \to 0^+}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) 
-oo
−∞-\infty 
= -8.86491367718932
 lim (cot(x)*log(x)*sin(x))
x->0-
lim⁡x→0−(log⁡(x)cot⁡(x)sin⁡(x))\lim_{x \to 0^-}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) 
-oo
−∞-\infty 
= (-8.85935629963627 + 3.13348227607372j)
= (-8.85935629963627 + 3.13348227607372j)
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(log⁡(x)cot⁡(x)sin⁡(x))=−∞\lim_{x \to 0^-}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) = -\infty
More at x→0 from the left
lim⁡x→0+(log⁡(x)cot⁡(x)sin⁡(x))=−∞\lim_{x \to 0^+}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) = -\infty
lim⁡x→∞(log⁡(x)cot⁡(x)sin⁡(x))\lim_{x \to \infty}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right)
More at x→oo
lim⁡x→1−(log⁡(x)cot⁡(x)sin⁡(x))=0\lim_{x \to 1^-}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) = 0
More at x→1 from the left
lim⁡x→1+(log⁡(x)cot⁡(x)sin⁡(x))=0\lim_{x \to 1^+}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right) = 0
More at x→1 from the right
lim⁡x→−∞(log⁡(x)cot⁡(x)sin⁡(x))\lim_{x \to -\infty}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right)
More at x→-oo
Numerical answer [src] 
-8.86491367718932
-8.86491367718932
The graph
Limit of the function cot(x)*log(x)*sin(x)
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