asymptotes of y=(3x^2-3x-2)/(x-1)

1) Let's find the asymptotes of

\frac{3 x^{2} - 3 x - 2}{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 found:

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

(3 x^{2} - 3 x - 2)

(x - 1)

⇥

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

⇥So the result of the euclidean division is:

3 x

The division lets us write

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

: the line to test is

y = 3 x

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

\frac{2}{- 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 = 3 x

; there is no horizontal asymptote

▶Asymptotes:

{x = 1, y = 3 x}

🖼/draw?f1=%283*x%5E2+-+3*x+-+2%29%2F%28x+-+1%29&v=x&x2=1&f3=3*x&b=-9.0&e=11.0&min=-33.0&max=39.0&dash=2,3