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