Limit of the function 1+x/2

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     /    x\
 lim |1 + -|
x->oo\    2/

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

Limit(1 + x/2, x, oo, dir='-')

Detail solution

Let's take the limit

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

Let's divide numerator and denominator by x:

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

=

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

Do Replacement

u = \frac{1}{x}

then

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

=

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

The final answer:

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

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

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

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

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

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

Rapid solution [src]

oo

\infty

Expand and simplify

The graph