asymptotes of f(x)=(x^2-4)/(x-1)

1) Let's find the asymptotes of

\frac{x^{2} - 4}{x - 1}

, 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 - 1 = 0

⇥ⓘ We add

1

to both sides

⇥Solution obtained:

x = 1

x

approaches

1

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 = 1

is therefore a vertical asymptote

⇥1.2) Perform polynomial long division of

(x^{2} - 4)

(x - 1)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{2} - 4$	$\frac{x^{2}}{x}$	$x$	$x \cdot (x - 1) = x^{2} - x$	$(x^{2} - 4) - (x^{2} - x) = x - 4$
$x - 4$	$\frac{x}{x}$	$1$	$x - 1 = x - 1$	$(x - 4) - (x - 1) = -3$

⇥So the result of the euclidean division is:

(x + 1)

The division lets us write

\frac{x^{2} - 4}{x - 1} = (x + 1) + \frac{3}{- x + 1}

: the line to test is

y = x + 1

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

\frac{3}{- x + 1}

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 + 1

; there is no horizontal asymptote

▶Asymptotes:

{x = 1, y = x + 1}

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