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sin(x)^4/x^4
Limit of the function/ sin(x)^4/x^4

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sin(x)^4/x^4

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sin(x)^4/(x^4)
 
sin(x)^((4/x)^4)
 
sin(x)^((4/x)^4)

Limit of the function sin(x)^4/x^4

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The solution

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     /   4   \
     |sin (x)|
 lim |-------|
x->0+|    4  |
     \   x   /
lim⁡x→0+(sin⁡4(x)x4)\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) 
Limit(sin(x)^4/(x^4), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+sin⁡4(x)=0\lim_{x \to 0^+} \sin^{4}{\left(x \right)} = 0
and limit for the denominator is
lim⁡x→0+x4=0\lim_{x \to 0^+} x^{4} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡4(x)x4)\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡4(x)x4)\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right)
=
lim⁡x→0+(ddxsin⁡4(x)ddxx4)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{4}{\left(x \right)}}{\frac{d}{d x} x^{4}}\right)
=
lim⁡x→0+(sin⁡3(x)cos⁡(x)x3)\lim_{x \to 0^+}\left(\frac{\sin^{3}{\left(x \right)} \cos{\left(x \right)}}{x^{3}}\right)
=
lim⁡x→0+(sin⁡3(x)x3)\lim_{x \to 0^+}\left(\frac{\sin^{3}{\left(x \right)}}{x^{3}}\right)
=
lim⁡x→0+(ddx4sin⁡3(x)ddx4x3)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 4 \sin^{3}{\left(x \right)}}{\frac{d}{d x} 4 x^{3}}\right)
=
lim⁡x→0+(sin⁡2(x)cos⁡(x)x2)\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}}\right)
=
lim⁡x→0+(sin⁡2(x)x2)\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)
=
lim⁡x→0+(ddx12sin⁡2(x)ddx12x2)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 12 \sin^{2}{\left(x \right)}}{\frac{d}{d x} 12 x^{2}}\right)
=
lim⁡x→0+(sin⁡(x)cos⁡(x)x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)
=
lim⁡x→0+(sin⁡(x)x)\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)
=
lim⁡x→0+(ddx24sin⁡(x)ddx24x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 24 \sin{\left(x \right)}}{\frac{d}{d x} 24 x}\right)
=
lim⁡x→0+cos⁡(x)\lim_{x \to 0^+} \cos{\left(x \right)}
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
11
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 4 time(s)
The graph
02468-8-6-4-2-101002
Plot the graph
One‐sided limits [src] 
     /   4   \
     |sin (x)|
 lim |-------|
x->0+|    4  |
     \   x   /
lim⁡x→0+(sin⁡4(x)x4)\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) 
1
11 
= 1
     /   4   \
     |sin (x)|
 lim |-------|
x->0-|    4  |
     \   x   /
lim⁡x→0−(sin⁡4(x)x4)\lim_{x \to 0^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) 
1
11 
= 1
= 1
Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡4(x)x4)=1\lim_{x \to 0^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 1
More at x→0 from the left
lim⁡x→0+(sin⁡4(x)x4)=1\lim_{x \to 0^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 1
lim⁡x→∞(sin⁡4(x)x4)=0\lim_{x \to \infty}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡4(x)x4)=sin⁡4(1)\lim_{x \to 1^-}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = \sin^{4}{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(sin⁡4(x)x4)=sin⁡4(1)\lim_{x \to 1^+}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = \sin^{4}{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(sin⁡4(x)x4)=0\lim_{x \to -\infty}\left(\frac{\sin^{4}{\left(x \right)}}{x^{4}}\right) = 0
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function sin(x)^4/x^4
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