Mister Exam

Other calculators:

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

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

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /-x + tan(x) \
 lim |------------|
x->0+\x + 2*sin(x)/
lim⁡x→0+(−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
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+(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.
lim⁡x→0+(−x+tan⁡(x)x+2sin⁡(x))\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)
=
lim⁡x→0+(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)
=
lim⁡x→0+(tan⁡2(x)2cos⁡(x)+1)\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{2 \cos{\left(x \right)} + 1}\right)
=
lim⁡x→0+(ddxtan⁡2(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)
=
lim⁡x→0+(−(2tan⁡2(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)
=
lim⁡x→0+(−tan⁡(x)sin⁡(x))\lim_{x \to 0^+}\left(- \frac{\tan{\left(x \right)}}{\sin{\left(x \right)}}\right)
=
lim⁡x→0+(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)
=
lim⁡x→0+(−tan⁡2(x)−1cos⁡(x))\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)
=
lim⁡x→0+(−tan⁡2(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
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
One‐sided limits [src] 
     /-x + tan(x) \
 lim |------------|
x->0+\x + 2*sin(x)/
lim⁡x→0+(−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)/
lim⁡x→0−(−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))
    © Mister Exam – Calculator