asymptotes of f(x)=(-x^3-5)/(x^2-4)

Let's find the asymptotes of

\frac{- x^{3} - 5}{x^{2} - 4}

, for real

x

Zeros of the denominator are candidates: an infinite limit confirms a vertical asymptote

Let's solve in

ℝ

x^{2} - 4 = 0

Solutions obtained:

x = {-2, 2}

-2

, the left-hand limit is

+∞

and the right-hand limit is

-∞

The limit is infinite:

x = -2

is a vertical asymptote

2

, the left-hand limit is

+∞

and the right-hand limit is

-∞

The limit is infinite:

x = 2

is a vertical asymptote

Compute the quotient and remainder of

- x^{3} - 5

x^{2} - 4

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$- x^{3} - 5$	$- \frac{x^{3}}{x^{2}}$	$- x$	$- x \cdot (x^{2} - 4) = - x^{3} + 4 x$	$(- x^{3} - 5) - (- x^{3} + 4 x) = - 4 x - 5$

So the result of the euclidean division is:

- x

The difference from

- x

- \frac{4 x + 5}{x^{2} - 4}

We check that the difference from

- x

tends to 0 at both infinities

The difference tends to

0

at +∞ and to

0

at -∞

The slant asymptote is

y = - x

; there is no horizontal asymptote

▶Asymptotes:

{x = -2, x = 2, y = - x}