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sin(x)/(10*x)
Limit of the function/ sin(x)/(10*x)

Limit of the function sin(x)/(10*x)

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     /sin(x)\
 lim |------|
x->0+\ 10*x /
lim⁡x→0+(sin⁡(x)10x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right) 
Limit(sin(x)/((10*x)), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(sin⁡(x)10x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right)
lim⁡x→0+(sin⁡(x)10x)=lim⁡u→0+(sin⁡(u)10u)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{10 u}\right)
=
lim⁡u→0+(sin⁡(u)u)10\frac{\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{10}
The limit
lim⁡u→0+(sin⁡(u)u)\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)
is first remarkable limit, is equal to 1.

The final answer:
lim⁡x→0+(sin⁡(x)10x)=110\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = \frac{1}{10}
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+(sin⁡(x)10)=0\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10}\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⁡(x)10x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡(x)10x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right)
=
lim⁡x→0+(ddxsin⁡(x)10ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(x \right)}}{10}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(cos⁡(x)10)\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{10}\right)
=
lim⁡x→0+110\lim_{x \to 0^+} \frac{1}{10}
=
lim⁡x→0+110\lim_{x \to 0^+} \frac{1}{10}
=
110\frac{1}{10}
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
02468-8-6-4-2-10100.2-0.1
Plot the graph
Rapid solution [src] 
1/10
110\frac{1}{10}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(x)10x)=110\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = \frac{1}{10}
More at x→0 from the left
lim⁡x→0+(sin⁡(x)10x)=110\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = \frac{1}{10}
lim⁡x→∞(sin⁡(x)10x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(x)10x)=sin⁡(1)10\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = \frac{\sin{\left(1 \right)}}{10}
More at x→1 from the left
lim⁡x→1+(sin⁡(x)10x)=sin⁡(1)10\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = \frac{\sin{\left(1 \right)}}{10}
More at x→1 from the right
lim⁡x→−∞(sin⁡(x)10x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{10 x}\right) = 0
More at x→-oo
One‐sided limits [src] 
     /sin(x)\
 lim |------|
x->0+\ 10*x /
lim⁡x→0+(sin⁡(x)10x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{10 x}\right) 
1/10
110\frac{1}{10} 
= 0.1
     /sin(x)\
 lim |------|
x->0-\ 10*x /
lim⁡x→0−(sin⁡(x)10x)\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)}}{10 x}\right) 
1/10
110\frac{1}{10} 
= 0.1
= 0.1
Numerical answer [src] 
0.1
0.1
The graph
Limit of the function sin(x)/(10*x)
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