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sin(sin(sin(x)))/x
Limit of the function/ sin(sin(sin(x)))/x

Limit of the function sin(sin(sin(x)))/x

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     /sin(sin(sin(x)))\
 lim |----------------|
x->0+\       x        /
lim⁡x→0+(sin⁡(sin⁡(sin⁡(x)))x)\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) 
Limit(sin(sin(sin(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⁡(sin⁡(sin⁡(x)))=0\lim_{x \to 0^+} \sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \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⁡(sin⁡(sin⁡(x)))x)\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right)
=
lim⁡x→0+(ddxsin⁡(sin⁡(sin⁡(x)))ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(cos⁡(x)cos⁡(sin⁡(x))cos⁡(sin⁡(sin⁡(x))))\lim_{x \to 0^+}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}\right)
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
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
02468-8-6-4-2-10102-1
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Rapid solution [src] 
1
11
Expand and simplify
One‐sided limits [src] 
     /sin(sin(sin(x)))\
 lim |----------------|
x->0+\       x        /
lim⁡x→0+(sin⁡(sin⁡(sin⁡(x)))x)\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) 
1
11 
= 1.0
     /sin(sin(sin(x)))\
 lim |----------------|
x->0-\       x        /
lim⁡x→0−(sin⁡(sin⁡(sin⁡(x)))x)\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) 
1
11 
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(sin⁡(sin⁡(x)))x)=1\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 1
More at x→0 from the left
lim⁡x→0+(sin⁡(sin⁡(sin⁡(x)))x)=1\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 1
lim⁡x→∞(sin⁡(sin⁡(sin⁡(x)))x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(sin⁡(sin⁡(x)))x)=sin⁡(sin⁡(sin⁡(1)))\lim_{x \to 1^-}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}
More at x→1 from the left
lim⁡x→1+(sin⁡(sin⁡(sin⁡(x)))x)=sin⁡(sin⁡(sin⁡(1)))\lim_{x \to 1^+}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}
More at x→1 from the right
lim⁡x→−∞(sin⁡(sin⁡(sin⁡(x)))x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)}}{x}\right) = 0
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function sin(sin(sin(x)))/x
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