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sqrt(x*sin(x))/x^2
Limit of the function/ sqrt(x*sin(x))/x^2

Limit of the function sqrt(x*sin(x))/x^2

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

i.e. limit for the numerator is
lim⁡x→0+xsin⁡(x)=0\lim_{x \to 0^+} \sqrt{x \sin{\left(x \right)}} = 0
and limit for the denominator is
lim⁡x→0+x2=0\lim_{x \to 0^+} x^{2} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(xsin⁡(x)x2)\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)
=
lim⁡x→0+(ddxxsin⁡(x)ddxx2)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{x \sin{\left(x \right)}}}{\frac{d}{d x} x^{2}}\right)
=
lim⁡x→0+(xsin⁡(x)(xcos⁡(x)2+sin⁡(x)2)2x2sin⁡(x))\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{2 x^{2} \sin{\left(x \right)}}\right)
=
lim⁡x→0+(ddx(xcos⁡(x)2+sin⁡(x)2)ddx2x2sin⁡(x)xsin⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{2 x^{2} \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)
=
lim⁡x→0+(−xsin⁡(x)2+cos⁡(x)x2cos⁡(x)xsin⁡(x)+3xsin⁡(x)xsin⁡(x))\lim_{x \to 0^+}\left(\frac{- \frac{x \sin{\left(x \right)}}{2} + \cos{\left(x \right)}}{\frac{x^{2} \cos{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}} + \frac{3 x \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)
=
lim⁡x→0+(−xsin⁡(x)2+cos⁡(x)x2cos⁡(x)xsin⁡(x)+3xsin⁡(x)xsin⁡(x))\lim_{x \to 0^+}\left(\frac{- \frac{x \sin{\left(x \right)}}{2} + \cos{\left(x \right)}}{\frac{x^{2} \cos{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}} + \frac{3 x \sin{\left(x \right)}}{\sqrt{x \sin{\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) 2 time(s)
The graph
02468-8-6-4-2-10100200
Plot the graph
One‐sided limits [src] 
     /  __________\
     |\/ x*sin(x) |
 lim |------------|
x->0+|      2     |
     \     x      /
lim⁡x→0+(xsin⁡(x)x2)\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) 
oo
∞\infty 
= 150.999448123822
     /  __________\
     |\/ x*sin(x) |
 lim |------------|
x->0-|      2     |
     \     x      /
lim⁡x→0−(xsin⁡(x)x2)\lim_{x \to 0^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) 
oo
∞\infty 
= 150.999448123822
= 150.999448123822
Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(xsin⁡(x)x2)=∞\lim_{x \to 0^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \infty
More at x→0 from the left
lim⁡x→0+(xsin⁡(x)x2)=∞\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \infty
lim⁡x→∞(xsin⁡(x)x2)\lim_{x \to \infty}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)
More at x→oo
lim⁡x→1−(xsin⁡(x)x2)=sin⁡(1)\lim_{x \to 1^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \sqrt{\sin{\left(1 \right)}}
More at x→1 from the left
lim⁡x→1+(xsin⁡(x)x2)=sin⁡(1)\lim_{x \to 1^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \sqrt{\sin{\left(1 \right)}}
More at x→1 from the right
lim⁡x→−∞(xsin⁡(x)x2)\lim_{x \to -\infty}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)
More at x→-oo
Numerical answer [src] 
150.999448123822
150.999448123822
The graph
Limit of the function sqrt(x*sin(x))/x^2
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