Mister Exam

Other calculators:

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

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

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

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

i.e. limit for the numerator is
lim⁡x→0+(−xsin⁡(x)−x+tan⁡(x))=0\lim_{x \to 0^+}\left(- x \sin{\left(x \right)} - x + \tan{\left(x \right)}\right) = 0
and limit for the denominator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(−sin⁡(x)+−x+tan⁡(x)x)\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(−xsin⁡(x)−x+tan⁡(x)x)\lim_{x \to 0^+}\left(\frac{- x \sin{\left(x \right)} - x + \tan{\left(x \right)}}{x}\right)
=
lim⁡x→0+(ddx(−xsin⁡(x)−x+tan⁡(x))ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \sin{\left(x \right)} - x + \tan{\left(x \right)}\right)}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(−xcos⁡(x)−sin⁡(x)+tan⁡2(x))\lim_{x \to 0^+}\left(- x \cos{\left(x \right)} - \sin{\left(x \right)} + \tan^{2}{\left(x \right)}\right)
=
lim⁡x→0+(−xcos⁡(x)−sin⁡(x)+tan⁡2(x))\lim_{x \to 0^+}\left(- x \cos{\left(x \right)} - \sin{\left(x \right)} + \tan^{2}{\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) 1 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 |-sin(x) + -----------|
x->0+\               x     /
lim⁡x→0+(−sin⁡(x)+−x+tan⁡(x)x)\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right) 
0
00 
= -1.04558111207684e-30
     /          -x + tan(x)\
 lim |-sin(x) + -----------|
x->0-\               x     /
lim⁡x→0−(−sin⁡(x)+−x+tan⁡(x)x)\lim_{x \to 0^-}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right) 
0
00 
= -1.13471551625255e-30
= -1.13471551625255e-30
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(−sin⁡(x)+−x+tan⁡(x)x)=0\lim_{x \to 0^-}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right) = 0
More at x→0 from the left
lim⁡x→0+(−sin⁡(x)+−x+tan⁡(x)x)=0\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right) = 0
lim⁡x→∞(−sin⁡(x)+−x+tan⁡(x)x)\lim_{x \to \infty}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right)
More at x→oo
lim⁡x→1−(−sin⁡(x)+−x+tan⁡(x)x)=−1−sin⁡(1)+tan⁡(1)\lim_{x \to 1^-}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right) = -1 - \sin{\left(1 \right)} + \tan{\left(1 \right)}
More at x→1 from the left
lim⁡x→1+(−sin⁡(x)+−x+tan⁡(x)x)=−1−sin⁡(1)+tan⁡(1)\lim_{x \to 1^+}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right) = -1 - \sin{\left(1 \right)} + \tan{\left(1 \right)}
More at x→1 from the right
lim⁡x→−∞(−sin⁡(x)+−x+tan⁡(x)x)\lim_{x \to -\infty}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right)
More at x→-oo
Numerical answer [src] 
-1.04558111207684e-30
-1.04558111207684e-30
The graph
Limit of the function -sin(x)+(-x+tan(x))/x
    © Mister Exam – Calculator