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Limit of the function (3+5*x)/(1+x)

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     /3 + 5*x\
 lim |-------|
x->oo\ 1 + x /

\lim_{x \to \infty}\left(\frac{5 x + 3}{x + 1}\right)

Limit((3 + 5*x)/(1 + x), x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty}\left(\frac{5 x + 3}{x + 1}\right)

Let's divide numerator and denominator by x:

\lim_{x \to \infty}\left(\frac{5 x + 3}{x + 1}\right)

\lim_{x \to \infty}\left(\frac{5 + \frac{3}{x}}{1 + \frac{1}{x}}\right)

Do Replacement

u = \frac{1}{x}

then

\lim_{x \to \infty}\left(\frac{5 + \frac{3}{x}}{1 + \frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{3 u + 5}{u + 1}\right)

\frac{0 \cdot 3 + 5}{1} = 5

The final answer:

\lim_{x \to \infty}\left(\frac{5 x + 3}{x + 1}\right) = 5

Lopital's rule

We have indeterminateness of type

oo/oo,

i.e. limit for the numerator is

\lim_{x \to \infty}\left(5 x + 3\right) = \infty

and limit for the denominator is

\lim_{x \to \infty}\left(x + 1\right) = \infty

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

\lim_{x \to \infty}\left(\frac{5 x + 3}{x + 1}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(5 x + 3\right)}{\frac{d}{d x} \left(x + 1\right)}\right)

\lim_{x \to \infty} 5

\lim_{x \to \infty} 5

5

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]

5

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{5 x + 3}{x + 1}\right) = 5

\lim_{x \to 0^-}\left(\frac{5 x + 3}{x + 1}\right) = 3

\lim_{x \to 0^+}\left(\frac{5 x + 3}{x + 1}\right) = 3

\lim_{x \to 1^-}\left(\frac{5 x + 3}{x + 1}\right) = 4

\lim_{x \to 1^+}\left(\frac{5 x + 3}{x + 1}\right) = 4

\lim_{x \to -\infty}\left(\frac{5 x + 3}{x + 1}\right) = 5

The graph