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x*cot(x/2)
Limit of the function/ x*cot(x/2)

Limit of the function x*cot(x/2)

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     /     /x\\
 lim |x*cot|-||
x->0+\     \2//
lim⁡x→0+(xcot⁡(x2))\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) 
Limit(x*cot(x/2), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
and limit for the denominator is
lim⁡x→0+1cot⁡(x2)=0\lim_{x \to 0^+} \frac{1}{\cot{\left(\frac{x}{2} \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(xcot⁡(x2))\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right)
=
lim⁡x→0+(ddxxddx1cot⁡(x2))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\cot{\left(\frac{x}{2} \right)}}}\right)
=
lim⁡x→0+(cot⁡2(x2)cot⁡2(x2)2+12)\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}\right)
=
lim⁡x→0+(cot⁡2(x2)cot⁡2(x2)2+12)\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}\right)
=
22
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
02468-8-6-4-2-1010-250250
Plot the graph
Rapid solution [src] 
2
22
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(xcot⁡(x2))=2\lim_{x \to 0^-}\left(x \cot{\left(\frac{x}{2} \right)}\right) = 2
More at x→0 from the left
lim⁡x→0+(xcot⁡(x2))=2\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) = 2
lim⁡x→∞(xcot⁡(x2))\lim_{x \to \infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)
More at x→oo
lim⁡x→1−(xcot⁡(x2))=1tan⁡(12)\lim_{x \to 1^-}\left(x \cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\tan{\left(\frac{1}{2} \right)}}
More at x→1 from the left
lim⁡x→1+(xcot⁡(x2))=1tan⁡(12)\lim_{x \to 1^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\tan{\left(\frac{1}{2} \right)}}
More at x→1 from the right
lim⁡x→−∞(xcot⁡(x2))\lim_{x \to -\infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)
More at x→-oo
One‐sided limits [src] 
     /     /x\\
 lim |x*cot|-||
x->0+\     \2//
lim⁡x→0+(xcot⁡(x2))\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) 
2
22 
= 2.0
     /     /x\\
 lim |x*cot|-||
x->0-\     \2//
lim⁡x→0−(xcot⁡(x2))\lim_{x \to 0^-}\left(x \cot{\left(\frac{x}{2} \right)}\right) 
2
22 
= 2.0
= 2.0
Numerical answer [src] 
2.0
2.0
The graph
Limit of the function x*cot(x/2)
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