asymptotes of (x^4-1)/x^3 — step-by-step solution

1) Let's find the asymptotes of

\frac{x^{4} - 1}{x^{3}}

, for real

x

Vertical asymptotes: find the excluded values, where the denominator is zero. Then check whether the curve goes to infinity near these values

⇥1.1) Let's solve in

ℝ

x^{3} = 0

⇥

x^{3} = 0

is equivalent to:

x = 0

⇥Solution found:

x = 0

x

approaches

0

from the left (smaller values), the function tends to

+∞

; from the right (larger values), it tends to

-∞

The function tends to infinity on at least one side: the line

x = 0

is therefore a vertical asymptote

⇥1.2) Perform polynomial long division of

(x^{4} - 1)

(x^{3})

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4} - 1$	$\frac{x^{4}}{x^{3}}$	$x$	$x x^{3} = x^{4}$	$(x^{4} - 1) - x^{4} = -1$

⇥So the result of the euclidean division is:

x

The division lets us write

\frac{x^{4} - 1}{x^{3}} = x - \frac{1}{x^{3}}

: the line to test is

y = x

For a slant asymptote, the difference between the function and the line must tend to 0: check the limit of

- \frac{1}{x^{3}}

at both infinities

The difference tends to

0

at +∞ and to

0

at -∞: the curve therefore approaches the line at both infinities

The slant asymptote is

y = x

; there is no horizontal asymptote

▶Asymptotes:

{x = 0, y = x}

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