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Limit of the function/ (2+n)/n

Limit of the function (2+n)/n

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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

Lopital's rule

We have indeterminateness of type

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

Plot the graph

Rapid solution [src]

1

Expand and simplify

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

\lim_{n \to 0^+}\left(\frac{n + 2}{n}\right) = \infty

\lim_{n \to 1^-}\left(\frac{n + 2}{n}\right) = 3

\lim_{n \to 1^+}\left(\frac{n + 2}{n}\right) = 3

\lim_{n \to -\infty}\left(\frac{n + 2}{n}\right) = 1

The graph