Mister Exam

Other calculators:

2*sin(4*x)/x
Limit of the function/ 2*sin(4*x)/x

Limit of the function 2*sin(4*x)/x

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /2*sin(4*x)\
 lim |----------|
x->0+\    x     /
lim⁡x→0+(2sin⁡(4x)x)\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) 
Limit(2*sin(4*x)/x, x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(2sin⁡(4x)x)\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)
Do replacement
u=4xu = 4 x
then
lim⁡x→0+(2sin⁡(4x)x)=lim⁡u→0+(8sin⁡(u)u)\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{8 \sin{\left(u \right)}}{u}\right)
=
8lim⁡u→0+(sin⁡(u)u)8 \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+(2sin⁡(4x)x)=8\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+(2sin⁡(4x))=0\lim_{x \to 0^+}\left(2 \sin{\left(4 x \right)}\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+(2sin⁡(4x)x)\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)
=
lim⁡x→0+(ddx2sin⁡(4x)ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 2 \sin{\left(4 x \right)}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(8cos⁡(4x))\lim_{x \to 0^+}\left(8 \cos{\left(4 x \right)}\right)
=
lim⁡x→0+8\lim_{x \to 0^+} 8
=
lim⁡x→0+8\lim_{x \to 0^+} 8
=
88
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
02468-8-6-4-2-1010-1010
Plot the graph
Rapid solution [src] 
8
88
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(2sin⁡(4x)x)=8\lim_{x \to 0^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8
More at x→0 from the left
lim⁡x→0+(2sin⁡(4x)x)=8\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8
lim⁡x→∞(2sin⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 0
More at x→oo
lim⁡x→1−(2sin⁡(4x)x)=2sin⁡(4)\lim_{x \to 1^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 2 \sin{\left(4 \right)}
More at x→1 from the left
lim⁡x→1+(2sin⁡(4x)x)=2sin⁡(4)\lim_{x \to 1^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 2 \sin{\left(4 \right)}
More at x→1 from the right
lim⁡x→−∞(2sin⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 0
More at x→-oo
One‐sided limits [src] 
     /2*sin(4*x)\
 lim |----------|
x->0+\    x     /
lim⁡x→0+(2sin⁡(4x)x)\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) 
8
88 
= 8
     /2*sin(4*x)\
 lim |----------|
x->0-\    x     /
lim⁡x→0−(2sin⁡(4x)x)\lim_{x \to 0^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) 
8
88 
= 8
= 8
Numerical answer [src] 
8.0
8.0
The graph
Limit of the function 2*sin(4*x)/x
    © Mister Exam – Calculator