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sin(x)*tan(x/2)
Limit of the function/ sin(x)*tan(x/2)

Limit of the function sin(x)*tan(x/2)

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

i.e. limit for the numerator is
lim⁡x→π+sin⁡(x)=0\lim_{x \to \pi^+} \sin{\left(x \right)} = 0
and limit for the denominator is
lim⁡x→π+1tan⁡(x2)=0\lim_{x \to \pi^+} \frac{1}{\tan{\left(\frac{x}{2} \right)}} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→π+(sin⁡(x)tan⁡(x2))\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)
=
lim⁡x→π+(ddxsin⁡(x)ddx1tan⁡(x2))\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\tan{\left(\frac{x}{2} \right)}}}\right)
=
lim⁡x→π+(cos⁡(x)tan⁡2(x2)−tan⁡2(x2)2−12)\lim_{x \to \pi^+}\left(\frac{\cos{\left(x \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{- \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}}\right)
=
lim⁡x→π+(cos⁡(x)tan⁡2(x2)−tan⁡2(x2)2−12)\lim_{x \to \pi^+}\left(\frac{\cos{\left(x \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{- \frac{\tan^{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
0123456-6-5-4-3-2-104
Plot the graph
One‐sided limits [src] 
      /          /x\\
 lim  |sin(x)*tan|-||
x->pi+\          \2//
lim⁡x→π+(sin⁡(x)tan⁡(x2))\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) 
2
22 
= 2.0
      /          /x\\
 lim  |sin(x)*tan|-||
x->pi-\          \2//
lim⁡x→π−(sin⁡(x)tan⁡(x2))\lim_{x \to \pi^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) 
2
22 
= 2.0
= 2.0
Rapid solution [src] 
2
22
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→π−(sin⁡(x)tan⁡(x2))=2\lim_{x \to \pi^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 2
More at x→pi from the left
lim⁡x→π+(sin⁡(x)tan⁡(x2))=2\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 2
lim⁡x→∞(sin⁡(x)tan⁡(x2))\lim_{x \to \infty}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)
More at x→oo
lim⁡x→0−(sin⁡(x)tan⁡(x2))=0\lim_{x \to 0^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 0
More at x→0 from the left
lim⁡x→0+(sin⁡(x)tan⁡(x2))=0\lim_{x \to 0^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 0
More at x→0 from the right
lim⁡x→1−(sin⁡(x)tan⁡(x2))=sin⁡(1)tan⁡(12)\lim_{x \to 1^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = \sin{\left(1 \right)} \tan{\left(\frac{1}{2} \right)}
More at x→1 from the left
lim⁡x→1+(sin⁡(x)tan⁡(x2))=sin⁡(1)tan⁡(12)\lim_{x \to 1^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = \sin{\left(1 \right)} \tan{\left(\frac{1}{2} \right)}
More at x→1 from the right
lim⁡x→−∞(sin⁡(x)tan⁡(x2))\lim_{x \to -\infty}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)
More at x→-oo
Numerical answer [src] 
2.0
2.0
The graph
Limit of the function sin(x)*tan(x/2)
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