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