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(-x+tan(x))/(x+2*sin(x))

Limit of the function (-x+tan(x))/(x+2*sin(x))

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     /-x + tan(x) \
 lim |------------|
x->0+\x + 2*sin(x)/
limx0+(x+tan(x)x+2sin(x))\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)
Limit((-x + tan(x))/(x + 2*sin(x)), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
limx0+(x+tan(x))=0\lim_{x \to 0^+}\left(- x + \tan{\left(x \right)}\right) = 0
and limit for the denominator is
limx0+(x+2sin(x))=0\lim_{x \to 0^+}\left(x + 2 \sin{\left(x \right)}\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
limx0+(x+tan(x)x+2sin(x))\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)
=
limx0+(ddx(x+tan(x))ddx(x+2sin(x)))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \tan{\left(x \right)}\right)}{\frac{d}{d x} \left(x + 2 \sin{\left(x \right)}\right)}\right)
=
limx0+(tan2(x)2cos(x)+1)\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{2 \cos{\left(x \right)} + 1}\right)
=
limx0+(ddxtan2(x)ddx(2cos(x)+1))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan^{2}{\left(x \right)}}{\frac{d}{d x} \left(2 \cos{\left(x \right)} + 1\right)}\right)
=
limx0+((2tan2(x)+2)tan(x)2sin(x))\lim_{x \to 0^+}\left(- \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{2 \sin{\left(x \right)}}\right)
=
limx0+(tan(x)sin(x))\lim_{x \to 0^+}\left(- \frac{\tan{\left(x \right)}}{\sin{\left(x \right)}}\right)
=
limx0+(ddx(tan(x))ddxsin(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \tan{\left(x \right)}\right)}{\frac{d}{d x} \sin{\left(x \right)}}\right)
=
limx0+(tan2(x)1cos(x))\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)
=
limx0+(tan2(x)1cos(x))\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)
=
00
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
02468-8-6-4-2-1010-2020
Rapid solution [src]
0
00
One‐sided limits [src]
     /-x + tan(x) \
 lim |------------|
x->0+\x + 2*sin(x)/
limx0+(x+tan(x)x+2sin(x))\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)
0
00
= -4.72324115407374e-31
     /-x + tan(x) \
 lim |------------|
x->0-\x + 2*sin(x)/
limx0(x+tan(x)x+2sin(x))\lim_{x \to 0^-}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)
0
00
= -4.72324115407374e-31
= -4.72324115407374e-31
Numerical answer [src]
-4.72324115407374e-31
-4.72324115407374e-31
The graph
Limit of the function (-x+tan(x))/(x+2*sin(x))