Mister Exam

Other calculators:

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

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

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /sin(sin(x))\
 lim |-----------|
x->0+\   sin(x)  /
lim⁡x→0+(sin⁡(sin⁡(x))sin⁡(x))\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) 
Limit(sin(sin(x))/sin(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⁡(x))=0\lim_{x \to 0^+} \sin{\left(\sin{\left(x \right)} \right)} = 0
and limit for the denominator is
lim⁡x→0+sin⁡(x)=0\lim_{x \to 0^+} \sin{\left(x \right)} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡(sin⁡(x))sin⁡(x))\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)
=
lim⁡x→0+(ddxsin⁡(sin⁡(x))ddxsin⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(\sin{\left(x \right)} \right)}}{\frac{d}{d x} \sin{\left(x \right)}}\right)
=
lim⁡x→0+cos⁡(sin⁡(x))\lim_{x \to 0^+} \cos{\left(\sin{\left(x \right)} \right)}
=
lim⁡x→0+1cos⁡(x)\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}
=
lim⁡x→0+1cos⁡(x)\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}
=
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
02468-8-6-4-2-10100.81.2
Plot the graph
One‐sided limits [src] 
     /sin(sin(x))\
 lim |-----------|
x->0+\   sin(x)  /
lim⁡x→0+(sin⁡(sin⁡(x))sin⁡(x))\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) 
1
11 
= 1.0
     /sin(sin(x))\
 lim |-----------|
x->0-\   sin(x)  /
lim⁡x→0−(sin⁡(sin⁡(x))sin⁡(x))\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) 
1
11 
= 1.0
= 1.0
Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(sin⁡(x))sin⁡(x))=1\lim_{x \to 0^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = 1
More at x→0 from the left
lim⁡x→0+(sin⁡(sin⁡(x))sin⁡(x))=1\lim_{x \to 0^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = 1
lim⁡x→∞(sin⁡(sin⁡(x))sin⁡(x))\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)
More at x→oo
lim⁡x→1−(sin⁡(sin⁡(x))sin⁡(x))=sin⁡(sin⁡(1))sin⁡(1)\lim_{x \to 1^-}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}
More at x→1 from the left
lim⁡x→1+(sin⁡(sin⁡(x))sin⁡(x))=sin⁡(sin⁡(1))sin⁡(1)\lim_{x \to 1^+}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right) = \frac{\sin{\left(\sin{\left(1 \right)} \right)}}{\sin{\left(1 \right)}}
More at x→1 from the right
lim⁡x→−∞(sin⁡(sin⁡(x))sin⁡(x))\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}}\right)
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function sin(sin(x))/sin(x)
    © Mister Exam – Calculator