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(-x+tan(x))/x^3
Limit of the function/ (-x+tan(x))/x^3

Limit of the function (-x+tan(x))/x^3

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

i.e. limit for the numerator is
lim⁡x→0+(−x+tan⁡(x))=0\lim_{x \to 0^+}\left(- x + \tan{\left(x \right)}\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+(−x+tan⁡(x)x3)\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)
=
lim⁡x→0+(ddx(−x+tan⁡(x))ddxx3)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \tan{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)
=
lim⁡x→0+(tan⁡2(x)3x2)\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{3 x^{2}}\right)
=
lim⁡x→0+(ddxtan⁡2(x)ddx3x2)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan^{2}{\left(x \right)}}{\frac{d}{d x} 3 x^{2}}\right)
=
lim⁡x→0+((2tan⁡2(x)+2)tan⁡(x)6x)\lim_{x \to 0^+}\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{6 x}\right)
=
lim⁡x→0+(tan⁡(x)3x)\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{3 x}\right)
=
lim⁡x→0+(tan⁡(x)3x)\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{3 x}\right)
=
13\frac{1}{3}
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
02468-8-6-4-2-10105-5
Plot the graph
Rapid solution [src] 
1/3
13\frac{1}{3}
Expand and simplify
One‐sided limits [src] 
     /-x + tan(x)\
 lim |-----------|
x->0+|      3    |
     \     x     /
lim⁡x→0+(−x+tan⁡(x)x3)\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) 
1/3
13\frac{1}{3} 
= 0.333333333333333
     /-x + tan(x)\
 lim |-----------|
x->0-|      3    |
     \     x     /
lim⁡x→0−(−x+tan⁡(x)x3)\lim_{x \to 0^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) 
1/3
13\frac{1}{3} 
= 0.333333333333333
= 0.333333333333333
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(−x+tan⁡(x)x3)=13\lim_{x \to 0^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = \frac{1}{3}
More at x→0 from the left
lim⁡x→0+(−x+tan⁡(x)x3)=13\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = \frac{1}{3}
lim⁡x→∞(−x+tan⁡(x)x3)\lim_{x \to \infty}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)
More at x→oo
lim⁡x→1−(−x+tan⁡(x)x3)=−1+tan⁡(1)\lim_{x \to 1^-}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = -1 + \tan{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(−x+tan⁡(x)x3)=−1+tan⁡(1)\lim_{x \to 1^+}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right) = -1 + \tan{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(−x+tan⁡(x)x3)\lim_{x \to -\infty}\left(\frac{- x + \tan{\left(x \right)}}{x^{3}}\right)
More at x→-oo
Numerical answer [src] 
0.333333333333333
0.333333333333333
The graph
Limit of the function (-x+tan(x))/x^3
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