Mister Exam
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How to use it?
- Limit of the function:
- Limit of x^x/factorial(2*x)
-
Limit of x*tan(4*x)/(1-cos(x))
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Limit of x*(-log(6+x)+log(x))
-
Limit of x/(3+7*x)
-
Identical expressions
- x*(-log(six +x)+log(x))
- x multiply by ( minus logarithm of (6 plus x) plus logarithm of (x))
- x multiply by ( minus logarithm of (six plus x) plus logarithm of (x))
- x(-log(6+x)+log(x))
- x-log6+x+logx
-
Similar expressions
- x*(-log(6+x)-log(x))
- x*(-log(6-x)+log(x))
- x*(log(6+x)+log(x))
Limit of the function x*(-log(6+x)+log(x))
The solution
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lim (x*(-log(6 + x) + log(x))) x->oo
$$\lim_{x \to \infty}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right)$$
Limit(x*(-log(6 + x) + log(x)), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} \frac{1}{\log{\left(x \right)} - \log{\left(x + 6 \right)}} = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\log{\left(x \right)} - \log{\left(x + 6 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} \log{\left(x + 6 \right)} + \log{\left(x + 6 \right)}^{2}}{\frac{1}{x + 6} - \frac{1}{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} \log{\left(x + 6 \right)} + \log{\left(x + 6 \right)}^{2}}{\frac{1}{x + 6} - \frac{1}{x}}\right)$$
=
$$-6$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
oo/-oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} \frac{1}{\log{\left(x \right)} - \log{\left(x + 6 \right)}} = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\log{\left(x \right)} - \log{\left(x + 6 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} \log{\left(x + 6 \right)} + \log{\left(x + 6 \right)}^{2}}{\frac{1}{x + 6} - \frac{1}{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2} - 2 \log{\left(x \right)} \log{\left(x + 6 \right)} + \log{\left(x + 6 \right)}^{2}}{\frac{1}{x + 6} - \frac{1}{x}}\right)$$
=
$$-6$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = -6$$
$$\lim_{x \to 0^-}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = - \log{\left(7 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = - \log{\left(7 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = -6$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = - \log{\left(7 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = - \log{\left(7 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \left(\log{\left(x \right)} - \log{\left(x + 6 \right)}\right)\right) = -6$$
More at x→-oo
The graph