Mister Exam
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How to use it?
- Limit of the function:
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Limit of sin(3*x)/(4*x)
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Limit of sqrt(x+sqrt(x))/(x^2+x^(1/3))^(1/4)
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Limit of cot(2*x)/cot(8*x)
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Limit of log(1+2^x)*log(1+3/x)
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Identical expressions
- log(one + two ^x)*log(one + three /x)
- logarithm of (1 plus 2 to the power of x) multiply by logarithm of (1 plus 3 divide by x)
- logarithm of (one plus two to the power of x) multiply by logarithm of (one plus three divide by x)
- log(1+2x)*log(1+3/x)
- log1+2x*log1+3/x
- log(1+2^x)log(1+3/x)
- log(1+2x)log(1+3/x)
- log1+2xlog1+3/x
- log1+2^xlog1+3/x
- log(1+2^x)*log(1+3 divide by x)
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Similar expressions
- log(1+2^x)*log(1-3/x)
- log(1-2^x)*log(1+3/x)
Limit of the function log(1+2^x)*log(1+3/x)
The solution
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/ / x\ / 3\\ lim |log\1 + 2 /*log|1 + -|| x->oo\ \ x//
$$\lim_{x \to \infty}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right)$$
Limit(log(1 + 2^x)*log(1 + 3/x), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} \log{\left(2^{x} + 1 \right)} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{x + 3}{x} \right)}} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\log{\left(\frac{x + 3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(2^{x} + 1 \right)}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{x + 3}{x} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{3 \left(- \frac{2^{x} x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}} - \frac{x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{3 \left(- \frac{2^{x} x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}} - \frac{x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}}\right)}\right)$$
=
$$3 \log{\left(2 \right)}$$
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
$$\lim_{x \to \infty} \log{\left(2^{x} + 1 \right)} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} \frac{1}{\log{\left(\frac{x + 3}{x} \right)}} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\log{\left(\frac{x + 3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(2^{x} + 1 \right)}}{\frac{d}{d x} \frac{1}{\log{\left(\frac{x + 3}{x} \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{3 \left(- \frac{2^{x} x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}} - \frac{x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{x^{2}}{3 \left(- \frac{2^{x} x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}} - \frac{x}{2^{x} x \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2} + 3 \cdot 2^{x} \log{\left(2 \right)} \log{\left(1 + \frac{3}{x} \right)}^{2}}\right)}\right)$$
=
$$3 \log{\left(2 \right)}$$
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
$$\lim_{x \to \infty}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 3 \log{\left(2 \right)}$$
$$\lim_{x \to 0^-}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 2 \log{\left(2 \right)} \log{\left(3 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 2 \log{\left(2 \right)} \log{\left(3 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 2 \log{\left(2 \right)} \log{\left(3 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 2 \log{\left(2 \right)} \log{\left(3 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\log{\left(1 + \frac{3}{x} \right)} \log{\left(2^{x} + 1 \right)}\right) = 0$$
More at x→-oo
The graph