Mister Exam

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Limit of the function (1+n)/(2+n)

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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

Lopital's rule

We have indeterminateness of type

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

Plot 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}

\lim_{n \to 0^+}\left(\frac{n + 1}{n + 2}\right) = \frac{1}{2}

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

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

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

Rapid solution [src]

1

Expand and simplify

The graph