Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (5+3*x)/(-5+x)
-
Limit of (-1+x)/log(x)
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Limit of (-tan(x)+sin(x))/x^3
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Limit of 7-2*x
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Identical expressions
- (-tan(x)+sin(x))/x^ three
- ( minus tangent of (x) plus sinus of (x)) divide by x cubed
- ( minus tangent of (x) plus sinus of (x)) divide by x to the power of three
- (-tan(x)+sin(x))/x3
- -tanx+sinx/x3
- (-tan(x)+sin(x))/x³
- (-tan(x)+sin(x))/x to the power of 3
- -tanx+sinx/x^3
- (-tan(x)+sin(x)) divide by x^3
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Similar expressions
- (-tan(x)-sin(x))/x^3
- (tan(x)+sin(x))/x^3
- (-tan(x)+sinx)/x^3
Limit of the function (-tan(x)+sin(x))/x^3
The solution
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[src]
/-tan(x) + sin(x)\
lim |----------------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
Limit((-tan(x) + sin(x))/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
$$\lim_{x \to 0^+} x^{3} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - \tan^{2}{\left(x \right)} - 1}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(x \right)} - \tan^{2}{\left(x \right)} - 1\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - 2 \tan^{3}{\left(x \right)} - 2 \tan{\left(x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} - 2 \tan^{3}{\left(x \right)} - 2 \tan{\left(x \right)}\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\cos{\left(x \right)}}{6} - \tan^{4}{\left(x \right)} - \frac{4 \tan^{2}{\left(x \right)}}{3} - \frac{1}{3}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\cos{\left(x \right)}}{6} - \tan^{4}{\left(x \right)} - \frac{4 \tan^{2}{\left(x \right)}}{3} - \frac{1}{3}\right)$$
=
$$- \frac{1}{2}$$
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
$$\lim_{x \to 0^+} x^{3} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \tan{\left(x \right)}\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)} - \tan^{2}{\left(x \right)} - 1}{3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(x \right)} - \tan^{2}{\left(x \right)} - 1\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - 2 \tan^{3}{\left(x \right)} - 2 \tan{\left(x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} - 2 \tan^{3}{\left(x \right)} - 2 \tan{\left(x \right)}\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\cos{\left(x \right)}}{6} - \tan^{4}{\left(x \right)} - \frac{4 \tan^{2}{\left(x \right)}}{3} - \frac{1}{3}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\cos{\left(x \right)}}{6} - \tan^{4}{\left(x \right)} - \frac{4 \tan^{2}{\left(x \right)}}{3} - \frac{1}{3}\right)$$
=
$$- \frac{1}{2}$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \tan{\left(1 \right)} + \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \tan{\left(1 \right)} + \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \tan{\left(1 \right)} + \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right) = - \tan{\left(1 \right)} + \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
More at x→-oo
One‐sided limits
[src]
/-tan(x) + sin(x)\
lim |----------------|
x->0+| 3 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
/-tan(x) + sin(x)\
lim |----------------|
x->0-| 3 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} - \tan{\left(x \right)}}{x^{3}}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
= -0.5
The graph