Mister Exam
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How to use it?
- Limit of the function:
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Limit of (-1+3*x)/(5+x^2+7*x)
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Limit of (7+x+x^2)/(-1+e^x)
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Limit of x^2/(-2+sqrt(4+x^2))
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Limit of (x^2-6*x)/(6+x^2-7*x)
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Identical expressions
- factorial(x)/(three ^x+x^ three)
- factorial(x) divide by (3 to the power of x plus x cubed )
- factorial(x) divide by (three to the power of x plus x to the power of three)
- factorial(x)/(3x+x3)
- factorialx/3x+x3
- factorial(x)/(3^x+x³)
- factorial(x)/(3 to the power of x+x to the power of 3)
- factorialx/3^x+x^3
- factorial(x) divide by (3^x+x^3)
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Similar expressions
- factorial(x)/(3^x-x^3)
Limit of the function factorial(x)/(3^x+x^3)
The solution
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/ x! \
lim |-------|
x->oo| x 3|
\3 + x /
$$\lim_{x \to \infty}\left(\frac{x!}{3^{x} + x^{3}}\right)$$
Limit(factorial(x)/(3^x + x^3), 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
$$\lim_{x \to \infty}\left(3^{x} + x^{3}\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x!}{3^{x} + x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} \left(3^{x} + x^{3}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{3^{x} \log{\left(3 \right)} + 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\frac{d}{d x} \left(3^{x} \log{\left(3 \right)} + 3 x^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{3^{x} \log{\left(3 \right)}^{2} + 6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{3^{x} \log{\left(3 \right)}^{2} + 6 x}\right)$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} x! = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(3^{x} + x^{3}\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x!}{3^{x} + x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} \left(3^{x} + x^{3}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{3^{x} \log{\left(3 \right)} + 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\frac{d}{d x} \left(3^{x} \log{\left(3 \right)} + 3 x^{2}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{3^{x} \log{\left(3 \right)}^{2} + 6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} + \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{3^{x} \log{\left(3 \right)}^{2} + 6 x}\right)$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x!}{3^{x} + x^{3}}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x!}{3^{x} + x^{3}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x!}{3^{x} + x^{3}}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x!}{3^{x} + x^{3}}\right) = \frac{1}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x!}{3^{x} + x^{3}}\right) = \frac{1}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x!}{3^{x} + x^{3}}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x!}{3^{x} + x^{3}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x!}{3^{x} + x^{3}}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x!}{3^{x} + x^{3}}\right) = \frac{1}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x!}{3^{x} + x^{3}}\right) = \frac{1}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x!}{3^{x} + x^{3}}\right) = 0$$
More at x→-oo
The graph