Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1+x)/(2+(-7+x)^(1/3))
-
Limit of (-1+x)*cos(1/(-1+x))
-
Limit of x/(1-(1+x)^(1/3))
-
Limit of tan(7*x)^2/sin(4*x^2)
- Graphing y =:
- log(sin(x))*sin(x)
-
Identical expressions
- log(sin(x))*sin(x)
- logarithm of ( sinus of (x)) multiply by sinus of (x)
- log(sin(x))sin(x)
- logsinxsinx
-
Similar expressions
- log(sinx)*sinx
Limit of the function log(sin(x))*sin(x)
The solution
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lim (log(sin(x))*sin(x)) x->0+
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right)$$
Limit(log(sin(x))*sin(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{1}{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2}}{4 \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}}{4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}}{4 \sin{\left(x \right)}}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} \cos{\left(x \right)}\right)}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2 \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{1}{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)^{2}}{4 \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}}{4 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{- \frac{\log{\left(\sin{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{2} - \frac{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{2}}{4 \sin{\left(x \right)}}\right)$$
=
$$0$$
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]
lim (log(sin(x))*sin(x)) x->0+
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right)$$
0
$$0$$
= -0.00185919843575947
lim (log(sin(x))*sin(x)) x->0-
$$\lim_{x \to 0^-}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right)$$
0
$$0$$
= (0.00188743473963812 - 0.000778278383302427j)
= (0.00188743473963812 - 0.000778278383302427j)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \left\langle -1, 1\right\rangle \log{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→oo
$$\lim_{x \to 1^-}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \log{\left(\sin{\left(1 \right)} \right)} \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \log{\left(\sin{\left(1 \right)} \right)} \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \left\langle -1, 1\right\rangle \log{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \left\langle -1, 1\right\rangle \log{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→oo
$$\lim_{x \to 1^-}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \log{\left(\sin{\left(1 \right)} \right)} \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \log{\left(\sin{\left(1 \right)} \right)} \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) = \left\langle -1, 1\right\rangle \log{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→-oo
The graph