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Limit of the function/ x^2/2

Limit of the function x^2/2

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

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

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

Detail solution

Let's take the limit

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

Let's divide numerator and denominator by x^2:

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

\lim_{x \to \infty} \frac{1}{2 \frac{1}{x^{2}}}

Do Replacement

u = \frac{1}{x}

then

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

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

The final answer:

\lim_{x \to \infty}\left(\frac{x^{2}}{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

Rapid solution [src]

oo

\infty

Expand and simplify

Other limits x→0, -oo, +oo, 1

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

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

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

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

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

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

The graph