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(1+2*n)/(-1+3*n)
Limit of the function/ (1+2*n)/(-1+3*n)

Limit of the function (1+2*n)/(-1+3*n)

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     /1 + 2*n \
 lim |--------|
n->oo\-1 + 3*n/
lim⁡n→∞(2n+13n−1)\lim_{n \to \infty}\left(\frac{2 n + 1}{3 n - 1}\right) 
Limit((1 + 2*n)/(-1 + 3*n), n, oo, dir='-')
Detail solution
Let's take the limit
lim⁡n→∞(2n+13n−1)\lim_{n \to \infty}\left(\frac{2 n + 1}{3 n - 1}\right)
Let's divide numerator and denominator by n:
lim⁡n→∞(2n+13n−1)\lim_{n \to \infty}\left(\frac{2 n + 1}{3 n - 1}\right) =
lim⁡n→∞(2+1n3−1n)\lim_{n \to \infty}\left(\frac{2 + \frac{1}{n}}{3 - \frac{1}{n}}\right)
Do Replacement
u=1nu = \frac{1}{n}
then
lim⁡n→∞(2+1n3−1n)=lim⁡u→0+(u+23−u)\lim_{n \to \infty}\left(\frac{2 + \frac{1}{n}}{3 - \frac{1}{n}}\right) = \lim_{u \to 0^+}\left(\frac{u + 2}{3 - u}\right)
=
23−0=23\frac{2}{3 - 0} = \frac{2}{3}

The final answer:
lim⁡n→∞(2n+13n−1)=23\lim_{n \to \infty}\left(\frac{2 n + 1}{3 n - 1}\right) = \frac{2}{3}
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡n→∞(2n+1)=∞\lim_{n \to \infty}\left(2 n + 1\right) = \infty
and limit for the denominator is
lim⁡n→∞(3n−1)=∞\lim_{n \to \infty}\left(3 n - 1\right) = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡n→∞(2n+13n−1)\lim_{n \to \infty}\left(\frac{2 n + 1}{3 n - 1}\right)
=
lim⁡n→∞(ddn(2n+1)ddn(3n−1))\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(2 n + 1\right)}{\frac{d}{d n} \left(3 n - 1\right)}\right)
=
lim⁡n→∞23\lim_{n \to \infty} \frac{2}{3}
=
lim⁡n→∞23\lim_{n \to \infty} \frac{2}{3}
=
23\frac{2}{3}
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
02468-8-6-4-2-1010-2525
Plot the graph
Rapid solution [src] 
2/3
23\frac{2}{3}
Expand and simplify
Other limits n→0, -oo, +oo, 1
lim⁡n→∞(2n+13n−1)=23\lim_{n \to \infty}\left(\frac{2 n + 1}{3 n - 1}\right) = \frac{2}{3}
lim⁡n→0−(2n+13n−1)=−1\lim_{n \to 0^-}\left(\frac{2 n + 1}{3 n - 1}\right) = -1
More at n→0 from the left
lim⁡n→0+(2n+13n−1)=−1\lim_{n \to 0^+}\left(\frac{2 n + 1}{3 n - 1}\right) = -1
More at n→0 from the right
lim⁡n→1−(2n+13n−1)=32\lim_{n \to 1^-}\left(\frac{2 n + 1}{3 n - 1}\right) = \frac{3}{2}
More at n→1 from the left
lim⁡n→1+(2n+13n−1)=32\lim_{n \to 1^+}\left(\frac{2 n + 1}{3 n - 1}\right) = \frac{3}{2}
More at n→1 from the right
lim⁡n→−∞(2n+13n−1)=23\lim_{n \to -\infty}\left(\frac{2 n + 1}{3 n - 1}\right) = \frac{2}{3}
More at n→-oo
The graph
Limit of the function (1+2*n)/(-1+3*n)
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