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

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

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     /sin(3*x)\
 lim |--------|
x->0+\  4*x   /
lim⁡x→0+(sin⁡(3x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) 
Limit(sin(3*x)/((4*x)), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(sin⁡(3x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right)
Do replacement
u=3xu = 3 x
then
lim⁡x→0+(sin⁡(3x)4x)=lim⁡u→0+(3sin⁡(u)4u)\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{4 u}\right)
=
3lim⁡u→0+(sin⁡(u)u)4\frac{3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{4}
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⁡(3x)4x)=34\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = \frac{3}{4}
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+sin⁡(3x)=0\lim_{x \to 0^+} \sin{\left(3 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⁡(3x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡(3x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right)
=
lim⁡x→0+(ddxsin⁡(3x)ddx4x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(3 x \right)}}{\frac{d}{d x} 4 x}\right)
=
lim⁡x→0+(3cos⁡(3x)4)\lim_{x \to 0^+}\left(\frac{3 \cos{\left(3 x \right)}}{4}\right)
=
lim⁡x→0+34\lim_{x \to 0^+} \frac{3}{4}
=
lim⁡x→0+34\lim_{x \to 0^+} \frac{3}{4}
=
34\frac{3}{4}
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
02468-8-6-4-2-10101-1
Plot the graph
Rapid solution [src] 
3/4
34\frac{3}{4}
Expand and simplify
One‐sided limits [src] 
     /sin(3*x)\
 lim |--------|
x->0+\  4*x   /
lim⁡x→0+(sin⁡(3x)4x)\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) 
3/4
34\frac{3}{4} 
= 0.75
     /sin(3*x)\
 lim |--------|
x->0-\  4*x   /
lim⁡x→0−(sin⁡(3x)4x)\lim_{x \to 0^-}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) 
3/4
34\frac{3}{4} 
= 0.75
= 0.75
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(3x)4x)=34\lim_{x \to 0^-}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = \frac{3}{4}
More at x→0 from the left
lim⁡x→0+(sin⁡(3x)4x)=34\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = \frac{3}{4}
lim⁡x→∞(sin⁡(3x)4x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(3x)4x)=sin⁡(3)4\lim_{x \to 1^-}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = \frac{\sin{\left(3 \right)}}{4}
More at x→1 from the left
lim⁡x→1+(sin⁡(3x)4x)=sin⁡(3)4\lim_{x \to 1^+}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = \frac{\sin{\left(3 \right)}}{4}
More at x→1 from the right
lim⁡x→−∞(sin⁡(3x)4x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{4 x}\right) = 0
More at x→-oo
Numerical answer [src] 
0.75
0.75
The graph
Limit of the function sin(3*x)/(4*x)
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