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Limit of the function x^2/(4-x^2)

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     /   2  \
     |  x   |
 lim |------|
x->oo|     2|
     \4 - x /

\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)

Limit(x^2/(4 - x^2), x, oo, dir='-')

Detail solution

Let's take the limit

\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)

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

\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)

\lim_{x \to \infty} \frac{1}{-1 + \frac{4}{x^{2}}}

Do Replacement

u = \frac{1}{x}

then

\lim_{x \to \infty} \frac{1}{-1 + \frac{4}{x^{2}}} = \lim_{u \to 0^+} \frac{1}{4 u^{2} - 1}

\frac{1}{-1 + 4 \cdot 0^{2}} = -1

The final answer:

\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1

Lopital's rule

We have indeterminateness of type

oo/-oo,

i.e. limit for the numerator is

\lim_{x \to \infty} x^{2} = \infty

and limit for the denominator is

\lim_{x \to \infty}\left(4 - x^{2}\right) = -\infty

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

\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} \left(4 - x^{2}\right)}\right)

\lim_{x \to \infty} -1

\lim_{x \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

-1

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1

\lim_{x \to 0^-}\left(\frac{x^{2}}{4 - x^{2}}\right) = 0

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

\lim_{x \to 1^-}\left(\frac{x^{2}}{4 - x^{2}}\right) = \frac{1}{3}

\lim_{x \to 1^+}\left(\frac{x^{2}}{4 - x^{2}}\right) = \frac{1}{3}

\lim_{x \to -\infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1

The graph