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sin(12*x)/(4*x)
Limit of the function/ sin(12*x)/(4*x)

Limit of the function sin(12*x)/(4*x)

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     /sin(12*x)\
 lim |---------|
x->0+\   4*x   /
lim⁡x→0+(sin⁡(12x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) 
Limit(sin(12*x)/((4*x)), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(sin⁡(12x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right)
Do replacement
u=12xu = 12 x
then
lim⁡x→0+(sin⁡(12x)4x)=lim⁡u→0+(3sin⁡(u)u)\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{u}\right)
=
3lim⁡u→0+(sin⁡(u)u)3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)
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⁡(12x)4x)=3\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = 3
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+sin⁡(12x)=0\lim_{x \to 0^+} \sin{\left(12 x \right)} = 0
and limit for the denominator is
lim⁡x→0+(4x)=0\lim_{x \to 0^+}\left(4 x\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡(12x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡(12x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right)
=
lim⁡x→0+(ddxsin⁡(12x)ddx4x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(12 x \right)}}{\frac{d}{d x} 4 x}\right)
=
lim⁡x→0+(3cos⁡(12x))\lim_{x \to 0^+}\left(3 \cos{\left(12 x \right)}\right)
=
lim⁡x→0+3\lim_{x \to 0^+} 3
=
lim⁡x→0+3\lim_{x \to 0^+} 3
=
33
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
02468-8-6-4-2-10105-5
Plot the graph
One‐sided limits [src] 
     /sin(12*x)\
 lim |---------|
x->0+\   4*x   /
lim⁡x→0+(sin⁡(12x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) 
3
33 
= 3.0
     /sin(12*x)\
 lim |---------|
x->0-\   4*x   /
lim⁡x→0−(sin⁡(12x)4x)\lim_{x \to 0^-}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) 
3
33 
= 3.0
= 3.0
Rapid solution [src] 
3
33
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(12x)4x)=3\lim_{x \to 0^-}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = 3
More at x→0 from the left
lim⁡x→0+(sin⁡(12x)4x)=3\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = 3
lim⁡x→∞(sin⁡(12x)4x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(12x)4x)=sin⁡(12)4\lim_{x \to 1^-}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = \frac{\sin{\left(12 \right)}}{4}
More at x→1 from the left
lim⁡x→1+(sin⁡(12x)4x)=sin⁡(12)4\lim_{x \to 1^+}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = \frac{\sin{\left(12 \right)}}{4}
More at x→1 from the right
lim⁡x→−∞(sin⁡(12x)4x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(12 x \right)}}{4 x}\right) = 0
More at x→-oo
Numerical answer [src] 
3.0
3.0
The graph
Limit of the function sin(12*x)/(4*x)
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