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sin(x^3)/x^3
Limit of the function/ sin(x^3)/x^3

Limit of the function sin(x^3)/x^3

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     /   / 3\\
     |sin\x /|
 lim |-------|
x->0+|    3  |
     \   x   /
lim⁡x→0+(sin⁡(x3)x3)\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) 
Limit(sin(x^3)/(x^3), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+sin⁡(x3)=0\lim_{x \to 0^+} \sin{\left(x^{3} \right)} = 0
and limit for the denominator is
lim⁡x→0+x3=0\lim_{x \to 0^+} x^{3} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(sin⁡(x3)x3)\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(sin⁡(x3)x3)\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right)
=
lim⁡x→0+(ddxsin⁡(x3)ddxx3)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x^{3} \right)}}{\frac{d}{d x} x^{3}}\right)
=
lim⁡x→0+cos⁡(x3)\lim_{x \to 0^+} \cos{\left(x^{3} \right)}
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
lim⁡x→0+1\lim_{x \to 0^+} 1
=
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) 3 time(s)
The graph
02468-8-6-4-2-10102-1
Plot the graph
Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sin⁡(x3)x3)=1\lim_{x \to 0^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 1
More at x→0 from the left
lim⁡x→0+(sin⁡(x3)x3)=1\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 1
lim⁡x→∞(sin⁡(x3)x3)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 0
More at x→oo
lim⁡x→1−(sin⁡(x3)x3)=sin⁡(1)\lim_{x \to 1^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = \sin{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(sin⁡(x3)x3)=sin⁡(1)\lim_{x \to 1^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = \sin{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(sin⁡(x3)x3)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) = 0
More at x→-oo
One‐sided limits [src] 
     /   / 3\\
     |sin\x /|
 lim |-------|
x->0+|    3  |
     \   x   /
lim⁡x→0+(sin⁡(x3)x3)\lim_{x \to 0^+}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) 
1
11 
= 1
     /   / 3\\
     |sin\x /|
 lim |-------|
x->0-|    3  |
     \   x   /
lim⁡x→0−(sin⁡(x3)x3)\lim_{x \to 0^-}\left(\frac{\sin{\left(x^{3} \right)}}{x^{3}}\right) 
1
11 
= 1
= 1
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function sin(x^3)/x^3
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