Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (x/(1+2*x))^x
- Limit of n2*(5/2+n/2)
-
Limit of (1-cos(4*x))/(1-cos(8*x))
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Limit of (1+x^2-4*x)/(1+2*x)
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Identical expressions
- (-x+tan(x))/(x+ two *sin(x))
- ( minus x plus tangent of (x)) divide by (x plus 2 multiply by sinus of (x))
- ( minus x plus tangent of (x)) divide by (x plus two multiply by sinus of (x))
- (-x+tan(x))/(x+2sin(x))
- -x+tanx/x+2sinx
- (-x+tan(x)) divide by (x+2*sin(x))
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Similar expressions
- (-x+tan(x))/(x-2*sin(x))
- (-x-tan(x))/(x+2*sin(x))
- (x+tan(x))/(x+2*sin(x))
- (-x+tan(x))/(x+2*sinx)
Limit of the function (-x+tan(x))/(x+2*sin(x))
The solution
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/-x + tan(x) \ lim |------------| x->0+\x + 2*sin(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
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x + \tan{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x + 2 \sin{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \tan{\left(x \right)}\right)}{\frac{d}{d x} \left(x + 2 \sin{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{2 \cos{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan^{2}{\left(x \right)}}{\frac{d}{d x} \left(2 \cos{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{2 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\tan{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \tan{\left(x \right)}\right)}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
=
$$0$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x + \tan{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x + 2 \sin{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x + \tan{\left(x \right)}\right)}{\frac{d}{d x} \left(x + 2 \sin{\left(x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan^{2}{\left(x \right)}}{2 \cos{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan^{2}{\left(x \right)}}{\frac{d}{d x} \left(2 \cos{\left(x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{2 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\tan{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \tan{\left(x \right)}\right)}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \tan^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
=
$$0$$
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
One‐sided limits
[src]
/-x + tan(x) \ lim |------------| x->0+\x + 2*sin(x)/
$$\lim_{x \to 0^+}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)$$
0
$$0$$
= -4.72324115407374e-31
/-x + tan(x) \ lim |------------| x->0-\x + 2*sin(x)/
$$\lim_{x \to 0^-}\left(\frac{- x + \tan{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\right)$$
0
$$0$$
= -4.72324115407374e-31
= -4.72324115407374e-31
The graph