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sin(4*x)/(3*x)
Limit of the function/ sin(4*x)/(3*x)

Limit of the function sin(4*x)/(3*x)

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     /sin(4*x)\
 lim |--------|
x->0+\  3*x   /
lim⁡x→0+(sin⁡(4x)3x)\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) 
Limit(sin(4*x)/((3*x)), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(sin⁡(4x)3x)\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right)
Do replacement
u=4xu = 4 x
then
lim⁡x→0+(sin⁡(4x)3x)=lim⁡u→0+(4sin⁡(u)3u)\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = \lim_{u \to 0^+}\left(\frac{4 \sin{\left(u \right)}}{3 u}\right)
=
4lim⁡u→0+(sin⁡(u)u)3\frac{4 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{3}
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⁡(4x)3x)=43\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = \frac{4}{3}
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+sin⁡(4x)=0\lim_{x \to 0^+} \sin{\left(4 x \right)} = 0
and limit for the denominator is
lim⁡x→0+(3x)=0\lim_{x \to 0^+}\left(3 x\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡(4x)3x)\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡(4x)3x)\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right)
=
lim⁡x→0+(ddxsin⁡(4x)ddx3x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(4 x \right)}}{\frac{d}{d x} 3 x}\right)
=
lim⁡x→0+(4cos⁡(4x)3)\lim_{x \to 0^+}\left(\frac{4 \cos{\left(4 x \right)}}{3}\right)
=
lim⁡x→0+43\lim_{x \to 0^+} \frac{4}{3}
=
lim⁡x→0+43\lim_{x \to 0^+} \frac{4}{3}
=
43\frac{4}{3}
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
02468-8-6-4-2-10102-2
Plot the graph
Rapid solution [src] 
4/3
43\frac{4}{3}
Expand and simplify
One‐sided limits [src] 
     /sin(4*x)\
 lim |--------|
x->0+\  3*x   /
lim⁡x→0+(sin⁡(4x)3x)\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) 
4/3
43\frac{4}{3} 
= 1.33333333333333
     /sin(4*x)\
 lim |--------|
x->0-\  3*x   /
lim⁡x→0−(sin⁡(4x)3x)\lim_{x \to 0^-}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) 
4/3
43\frac{4}{3} 
= 1.33333333333333
= 1.33333333333333
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(4x)3x)=43\lim_{x \to 0^-}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = \frac{4}{3}
More at x→0 from the left
lim⁡x→0+(sin⁡(4x)3x)=43\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = \frac{4}{3}
lim⁡x→∞(sin⁡(4x)3x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(4x)3x)=sin⁡(4)3\lim_{x \to 1^-}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = \frac{\sin{\left(4 \right)}}{3}
More at x→1 from the left
lim⁡x→1+(sin⁡(4x)3x)=sin⁡(4)3\lim_{x \to 1^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = \frac{\sin{\left(4 \right)}}{3}
More at x→1 from the right
lim⁡x→−∞(sin⁡(4x)3x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) = 0
More at x→-oo
Numerical answer [src] 
1.33333333333333
1.33333333333333
The graph
Limit of the function sin(4*x)/(3*x)
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