Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (2+x^2+3*x)/(2+2*x^2+5*x)
-
Limit of (asin(x)^2-atan(x)^2+sin(2*x))/(3*x)
-
Limit of 1+(4/3)^n
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Limit of n/(-1+n)
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Identical expressions
- n/(- one +n)
- n divide by ( minus 1 plus n)
- n divide by ( minus one plus n)
- n/-1+n
- n divide by (-1+n)
-
Similar expressions
- n/(-1-n)
- n/(1+n)
- cos(n)/(-1+n)
- n/(-1+n^2)
- (1+n^3-n)/(-1+n^2+2*n)
- (sqrt(n)-n)/(-1+n^(2/3))
Limit of the function n/(-1+n)
The solution
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/ n \ lim |------| n->oo\-1 + n/
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$
Limit(n/(-1 + n), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$ =
$$\lim_{n \to \infty} \frac{1}{1 - \frac{1}{n}}$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty} \frac{1}{1 - \frac{1}{n}} = \lim_{u \to 0^+} \frac{1}{1 - u}$$
=
$$\frac{1}{1 - 0} = 1$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right) = 1$$
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$ =
$$\lim_{n \to \infty} \frac{1}{1 - \frac{1}{n}}$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty} \frac{1}{1 - \frac{1}{n}} = \lim_{u \to 0^+} \frac{1}{1 - u}$$
=
$$\frac{1}{1 - 0} = 1$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right) = 1$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty} n = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(n - 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n}{\frac{d}{d n} \left(n - 1\right)}\right)$$
=
$$\lim_{n \to \infty} 1$$
=
$$\lim_{n \to \infty} 1$$
=
$$1$$
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
$$\lim_{n \to \infty} n = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(n - 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n}{\frac{d}{d n} \left(n - 1\right)}\right)$$
=
$$\lim_{n \to \infty} 1$$
=
$$\lim_{n \to \infty} 1$$
=
$$1$$
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 n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{n}{n - 1}\right) = 1$$
$$\lim_{n \to 0^-}\left(\frac{n}{n - 1}\right) = 0$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n}{n - 1}\right) = 0$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n}{n - 1}\right) = -\infty$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n}{n - 1}\right) = \infty$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n}{n - 1}\right) = 1$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{n}{n - 1}\right) = 0$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{n}{n - 1}\right) = 0$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{n}{n - 1}\right) = -\infty$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{n}{n - 1}\right) = \infty$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{n}{n - 1}\right) = 1$$
More at n→-oo
The graph