Mister Exam
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How to use it?
- Limit of the function:
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Limit of (2+n)/n
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Limit of exp(1/x)
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Limit of 1-cos(x)/x^2
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Limit of log(2*x)*log(-1+2*x)
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Identical expressions
- log(two *x)*log(- one + two *x)
- logarithm of (2 multiply by x) multiply by logarithm of ( minus 1 plus 2 multiply by x)
- logarithm of (two multiply by x) multiply by logarithm of ( minus one plus two multiply by x)
- log(2x)log(-1+2x)
- log2xlog-1+2x
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Similar expressions
- log(2*x)*log(-1-2*x)
- log(2*x)*log(1+2*x)
Limit of the function log(2*x)*log(-1+2*x)
The solution
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[src]
lim (log(2*x)*log(-1 + 2*x)) x->1/2+
$$\lim_{x \to \frac{1}{2}^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right)$$
Limit(log(2*x)*log(-1 + 2*x), x, 1/2)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \frac{1}{2}^+} \log{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to \frac{1}{2}^+} \frac{1}{\log{\left(2 x - 1 \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \frac{1}{2}^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \log{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\log{\left(2 x - 1 \right)}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(- \frac{\left(2 x - 1\right) \log{\left(2 x - 1 \right)}^{2}}{2 x}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(- \frac{2 x - 1}{2 x}\right)}{\frac{d}{d x} \frac{1}{\log{\left(2 x - 1 \right)}^{2}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(- \frac{\left(- \frac{1}{x} + \frac{2 x - 1}{2 x^{2}}\right) \left(2 x - 1\right) \log{\left(2 x - 1 \right)}^{3}}{4}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(- \frac{\left(2 x - 1\right) \log{\left(2 x - 1 \right)}^{3}}{4}\right)}{\frac{d}{d x} \frac{1}{- \frac{1}{x} + \frac{2 x - 1}{2 x^{2}}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\left(- \frac{1}{x} + \frac{2 x - 1}{2 x^{2}}\right)^{2} \left(- \frac{\log{\left(2 x - 1 \right)}^{3}}{2} - \frac{3 \log{\left(2 x - 1 \right)}^{2}}{2}\right)}{- \frac{2}{x^{2}} + \frac{2 x - 1}{x^{3}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\log{\left(2 x - 1 \right)}^{3}}{4} + \frac{3 \log{\left(2 x - 1 \right)}^{2}}{4}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\log{\left(2 x - 1 \right)}^{3}}{4} + \frac{3 \log{\left(2 x - 1 \right)}^{2}}{4}\right)$$
=
$$0$$
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
$$\lim_{x \to \frac{1}{2}^+} \log{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to \frac{1}{2}^+} \frac{1}{\log{\left(2 x - 1 \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \frac{1}{2}^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \log{\left(2 x \right)}}{\frac{d}{d x} \frac{1}{\log{\left(2 x - 1 \right)}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(- \frac{\left(2 x - 1\right) \log{\left(2 x - 1 \right)}^{2}}{2 x}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(- \frac{2 x - 1}{2 x}\right)}{\frac{d}{d x} \frac{1}{\log{\left(2 x - 1 \right)}^{2}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(- \frac{\left(- \frac{1}{x} + \frac{2 x - 1}{2 x^{2}}\right) \left(2 x - 1\right) \log{\left(2 x - 1 \right)}^{3}}{4}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(- \frac{\left(2 x - 1\right) \log{\left(2 x - 1 \right)}^{3}}{4}\right)}{\frac{d}{d x} \frac{1}{- \frac{1}{x} + \frac{2 x - 1}{2 x^{2}}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\left(- \frac{1}{x} + \frac{2 x - 1}{2 x^{2}}\right)^{2} \left(- \frac{\log{\left(2 x - 1 \right)}^{3}}{2} - \frac{3 \log{\left(2 x - 1 \right)}^{2}}{2}\right)}{- \frac{2}{x^{2}} + \frac{2 x - 1}{x^{3}}}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\log{\left(2 x - 1 \right)}^{3}}{4} + \frac{3 \log{\left(2 x - 1 \right)}^{2}}{4}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{\log{\left(2 x - 1 \right)}^{3}}{4} + \frac{3 \log{\left(2 x - 1 \right)}^{2}}{4}\right)$$
=
$$0$$
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(2*x)*log(-1 + 2*x)) x->1/2+
$$\lim_{x \to \frac{1}{2}^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right)$$
0
$$0$$
= -0.00315612534445693
lim (log(2*x)*log(-1 + 2*x)) x->1/2-
$$\lim_{x \to \frac{1}{2}^-}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right)$$
0
$$0$$
= (0.000515070744372563 + 1.00253195903717e-44j)
= (0.000515070744372563 + 1.00253195903717e-44j)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \frac{1}{2}^-}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
More at x→1/2 from the left
$$\lim_{x \to \frac{1}{2}^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = - \infty i$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = - \infty i$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = \infty$$
More at x→-oo
More at x→1/2 from the left
$$\lim_{x \to \frac{1}{2}^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = - \infty i$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = - \infty i$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\log{\left(2 x \right)} \log{\left(2 x - 1 \right)}\right) = \infty$$
More at x→-oo
The graph