Mister Exam

Lang:

Limit of the function (6+n)/(4+n)

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     /6 + n\
 lim |-----|
n->oo\4 + n/

\lim_{n \to \infty}\left(\frac{n + 6}{n + 4}\right)

Limit((6 + n)/(4 + n), n, oo, dir='-')

Detail solution

Let's take the limit

\lim_{n \to \infty}\left(\frac{n + 6}{n + 4}\right)

Let's divide numerator and denominator by n:

\lim_{n \to \infty}\left(\frac{n + 6}{n + 4}\right)

\lim_{n \to \infty}\left(\frac{1 + \frac{6}{n}}{1 + \frac{4}{n}}\right)

Do Replacement

u = \frac{1}{n}

then

\lim_{n \to \infty}\left(\frac{1 + \frac{6}{n}}{1 + \frac{4}{n}}\right) = \lim_{u \to 0^+}\left(\frac{6 u + 1}{4 u + 1}\right)

\frac{0 \cdot 6 + 1}{0 \cdot 4 + 1} = 1

The final answer:

\lim_{n \to \infty}\left(\frac{n + 6}{n + 4}\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 + 6\right) = \infty

and limit for the denominator is

\lim_{n \to \infty}\left(n + 4\right) = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{n \to \infty}\left(\frac{n + 6}{n + 4}\right)

\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(n + 6\right)}{\frac{d}{d n} \left(n + 4\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

Rapid solution [src]

1

Expand and simplify

Other limits n→0, -oo, +oo, 1

\lim_{n \to \infty}\left(\frac{n + 6}{n + 4}\right) = 1

\lim_{n \to 0^-}\left(\frac{n + 6}{n + 4}\right) = \frac{3}{2}

\lim_{n \to 0^+}\left(\frac{n + 6}{n + 4}\right) = \frac{3}{2}

\lim_{n \to 1^-}\left(\frac{n + 6}{n + 4}\right) = \frac{7}{5}

\lim_{n \to 1^+}\left(\frac{n + 6}{n + 4}\right) = \frac{7}{5}

\lim_{n \to -\infty}\left(\frac{n + 6}{n + 4}\right) = 1

The graph