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