Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (2*x/(-3+2*x))^(3*x)
-
Limit of (1-cos(8*x))/x^2
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Limit of (-sin(x)+tan(x))/sin(x)^3
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Limit of (-2+x^2-x)/(-2+x)
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Identical expressions
- (-sin(x)+tan(x))/sin(x)^ three
- ( minus sinus of (x) plus tangent of (x)) divide by sinus of (x) cubed
- ( minus sinus of (x) plus tangent of (x)) divide by sinus of (x) to the power of three
- (-sin(x)+tan(x))/sin(x)3
- -sinx+tanx/sinx3
- (-sin(x)+tan(x))/sin(x)³
- (-sin(x)+tan(x))/sin(x) to the power of 3
- -sinx+tanx/sinx^3
- (-sin(x)+tan(x)) divide by sin(x)^3
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Similar expressions
- (sin(x)+tan(x))/sin(x)^3
- (-sin(x)-tan(x))/sin(x)^3
- (-sinx+tan(x))/sinx^3
Limit of the function (-sin(x)+tan(x))/sin(x)^3
The solution
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[src]
/-sin(x) + tan(x)\
lim |----------------|
x->0+| 3 |
\ sin (x) /
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
Limit((-sin(x) + tan(x))/sin(x)^3, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin^{3}{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right)}{\frac{d}{d x} \sin^{3}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}{3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}{3 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{\frac{d}{d x} 3 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} + \sin{\left(x \right)}}{6 \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + 2 \tan^{3}{\left(x \right)} + 2 \tan{\left(x \right)}}{6 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} + 2 \tan^{3}{\left(x \right)} + 2 \tan{\left(x \right)}\right)}{\frac{d}{d x} 6 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + \cos{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 2}{6 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{6} + \tan^{4}{\left(x \right)} + \frac{4 \tan^{2}{\left(x \right)}}{3} + \frac{1}{3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{6} + \tan^{4}{\left(x \right)} + \frac{4 \tan^{2}{\left(x \right)}}{3} + \frac{1}{3}\right)$$
=
$$\frac{1}{2}$$
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
$$\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin^{3}{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} + \tan{\left(x \right)}\right)}{\frac{d}{d x} \sin^{3}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}{3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1}{3 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \cos{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{\frac{d}{d x} 3 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} + \sin{\left(x \right)}}{6 \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + 2 \tan^{3}{\left(x \right)} + 2 \tan{\left(x \right)}}{6 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} + 2 \tan^{3}{\left(x \right)} + 2 \tan{\left(x \right)}\right)}{\frac{d}{d x} 6 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} + \cos{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 2}{6 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{6} + \tan^{4}{\left(x \right)} + \frac{4 \tan^{2}{\left(x \right)}}{3} + \frac{1}{3}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{6} + \tan^{4}{\left(x \right)} + \frac{4 \tan^{2}{\left(x \right)}}{3} + \frac{1}{3}\right)$$
=
$$\frac{1}{2}$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = - \frac{- \tan{\left(1 \right)} + \sin{\left(1 \right)}}{\sin^{3}{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = - \frac{- \tan{\left(1 \right)} + \sin{\left(1 \right)}}{\sin^{3}{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = - \frac{- \tan{\left(1 \right)} + \sin{\left(1 \right)}}{\sin^{3}{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right) = - \frac{- \tan{\left(1 \right)} + \sin{\left(1 \right)}}{\sin^{3}{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
More at x→-oo
One‐sided limits
[src]
/-sin(x) + tan(x)\
lim |----------------|
x->0+| 3 |
\ sin (x) /
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/-sin(x) + tan(x)\
lim |----------------|
x->0-| 3 |
\ sin (x) /
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} + \tan{\left(x \right)}}{\sin^{3}{\left(x \right)}}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
The graph