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asin(x)*cot(x)
Limit of the function/ asin(x)*cot(x)

Limit of the function asin(x)*cot(x)

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

i.e. limit for the numerator is
lim⁡x→0+asin⁡(x)=0\lim_{x \to 0^+} \operatorname{asin}{\left(x \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+(cot⁡(x)asin⁡(x))\lim_{x \to 0^+}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right)
=
lim⁡x→0+(ddxasin⁡(x)ddx1cot⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(x \right)}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)
=
lim⁡x→0+(11−x2(1+1cot⁡2(x)))\lim_{x \to 0^+}\left(\frac{1}{\sqrt{1 - x^{2}} \left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right)}\right)
=
lim⁡x→0+(11−x2(1+1cot⁡2(x)))\lim_{x \to 0^+}\left(\frac{1}{\sqrt{1 - x^{2}} \left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right)}\right)
=
11
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
02468-8-6-4-2-10100.81.1
Plot the graph
One‐sided limits [src] 
 lim (asin(x)*cot(x))
x->0+
lim⁡x→0+(cot⁡(x)asin⁡(x))\lim_{x \to 0^+}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right) 
1
11 
= 1.0
 lim (asin(x)*cot(x))
x->0-
lim⁡x→0−(cot⁡(x)asin⁡(x))\lim_{x \to 0^-}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right) 
1
11 
= 1.0
= 1.0
Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(cot⁡(x)asin⁡(x))=1\lim_{x \to 0^-}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right) = 1
More at x→0 from the left
lim⁡x→0+(cot⁡(x)asin⁡(x))=1\lim_{x \to 0^+}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right) = 1
lim⁡x→∞(cot⁡(x)asin⁡(x))\lim_{x \to \infty}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right)
More at x→oo
lim⁡x→1−(cot⁡(x)asin⁡(x))=π2tan⁡(1)\lim_{x \to 1^-}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right) = \frac{\pi}{2 \tan{\left(1 \right)}}
More at x→1 from the left
lim⁡x→1+(cot⁡(x)asin⁡(x))=π2tan⁡(1)\lim_{x \to 1^+}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right) = \frac{\pi}{2 \tan{\left(1 \right)}}
More at x→1 from the right
lim⁡x→−∞(cot⁡(x)asin⁡(x))\lim_{x \to -\infty}\left(\cot{\left(x \right)} \operatorname{asin}{\left(x \right)}\right)
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function asin(x)*cot(x)
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