Mister Exam
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How to use it?
- Limit of the function:
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Limit of e^(x/2)
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Limit of (1+x)^3-(-1+x)^3/(1+x^2)
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Limit of (-24+x^2+2*x)/(-4+x)
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Limit of (1+x^2-2*x)/(1+x^3)
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Identical expressions
- y*factorial(x)/factorial(one +x)
- y multiply by factorial(x) divide by factorial(1 plus x)
- y multiply by factorial(x) divide by factorial(one plus x)
- yfactorial(x)/factorial(1+x)
- yfactorialx/factorial1+x
- y*factorial(x) divide by factorial(1+x)
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Similar expressions
- y*factorial(x)/factorial(1-x)
Limit of the function y*factorial(x)/factorial(1+x)
The solution
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/ y*x! \ lim |--------| x->oo\(1 + x)!/
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
Limit((y*factorial(x))/factorial(1 + x), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} \frac{1}{\left(x + 1\right)!} = 0$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(\frac{1}{y x!}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 0 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to \infty} \frac{1}{\left(x + 1\right)!} = 0$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(\frac{1}{y x!}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right)$$
=
$$0$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 0 time(s)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{y x!}{\left(x + 1\right)!}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{y x!}{\left(x + 1\right)!}\right) = y$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{y x!}{\left(x + 1\right)!}\right) = y$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{y x!}{\left(x + 1\right)!}\right) = \frac{y}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{y x!}{\left(x + 1\right)!}\right) = \frac{y}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{y x!}{\left(x + 1\right)!}\right) = y$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{y x!}{\left(x + 1\right)!}\right) = y$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{y x!}{\left(x + 1\right)!}\right) = y$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{y x!}{\left(x + 1\right)!}\right) = \frac{y}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{y x!}{\left(x + 1\right)!}\right) = \frac{y}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{y x!}{\left(x + 1\right)!}\right) = y$$
More at x→-oo