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

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

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

i.e. limit for the numerator is
lim⁡x→0+sin⁡(2x)=0\lim_{x \to 0^+} \sqrt{\sin{\left(2 x \right)}} = 0
and limit for the denominator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡(2x)x)\lim_{x \to 0^+}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right)
=
lim⁡x→0+(ddxsin⁡(2x)ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{\sin{\left(2 x \right)}}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(cos⁡(2x)sin⁡(2x))\lim_{x \to 0^+}\left(\frac{\cos{\left(2 x \right)}}{\sqrt{\sin{\left(2 x \right)}}}\right)
=
lim⁡x→0+1sin⁡(2x)\lim_{x \to 0^+} \frac{1}{\sqrt{\sin{\left(2 x \right)}}}
=
lim⁡x→0+1sin⁡(2x)\lim_{x \to 0^+} \frac{1}{\sqrt{\sin{\left(2 x \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
02468-8-6-4-2-1010-2020
Plot the graph
Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(2x)x)=∞\lim_{x \to 0^-}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = \infty
More at x→0 from the left
lim⁡x→0+(sin⁡(2x)x)=∞\lim_{x \to 0^+}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = \infty
lim⁡x→∞(sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(2x)x)=sin⁡(2)\lim_{x \to 1^-}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = \sqrt{\sin{\left(2 \right)}}
More at x→1 from the left
lim⁡x→1+(sin⁡(2x)x)=sin⁡(2)\lim_{x \to 1^+}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = \sqrt{\sin{\left(2 \right)}}
More at x→1 from the right
lim⁡x→−∞(sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) = 0
More at x→-oo
One‐sided limits [src] 
     /  __________\
     |\/ sin(2*x) |
 lim |------------|
x->0+\     x      /
lim⁡x→0+(sin⁡(2x)x)\lim_{x \to 0^+}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) 
oo
∞\infty 
= 17.3778931420175
     /  __________\
     |\/ sin(2*x) |
 lim |------------|
x->0-\     x      /
lim⁡x→0−(sin⁡(2x)x)\lim_{x \to 0^-}\left(\frac{\sqrt{\sin{\left(2 x \right)}}}{x}\right) 
-oo*I
−∞i- \infty i 
= (0.0 - 17.3778931420175j)
= (0.0 - 17.3778931420175j)
Numerical answer [src] 
17.3778931420175
17.3778931420175
The graph
Limit of the function sqrt(sin(2*x))/x
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