Mister Exam

Lang:

Limit of the function/ 2+x

Limit of the function 2+x

You have entered [src]

 lim (2 + x)
x->oo

\lim_{x \to \infty}\left(x + 2\right)

Limit(2 + x, x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty}\left(x + 2\right)

Let's divide numerator and denominator by x:

\lim_{x \to \infty}\left(x + 2\right)

\lim_{x \to \infty}\left(\frac{1 + \frac{2}{x}}{\frac{1}{x}}\right)

Do Replacement

u = \frac{1}{x}

then

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

\frac{0 \cdot 2 + 1}{0} = \infty

The final answer:

\lim_{x \to \infty}\left(x + 2\right) = \infty

Lopital's rule

There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type

The graph

Plot the graph

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(x + 2\right) = \infty

\lim_{x \to 0^-}\left(x + 2\right) = 2

\lim_{x \to 0^+}\left(x + 2\right) = 2

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

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

\lim_{x \to -\infty}\left(x + 2\right) = -\infty

Rapid solution [src]

oo

\infty

Expand and simplify

The graph