asymptotes of y=(x+2)/(x+4) — step-by-step solution

1) Let's find the asymptotes of

\frac{x + 2}{x + 4}

, 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 + 4 = 0

⇥ⓘ We subtract

4

from both sides

⇥Solution found:

x = -4

x

approaches

-4

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 = -4

is therefore a vertical asymptote

⇥1.2) Compute the quotient and remainder of

(x + 2)

(x + 4)

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x + 2$	$\frac{x}{x}$	$1$	$x + 4 = x + 4$	$(x + 2) - (x + 4) = -2$

⇥So the result of the euclidean division is:

1

The quotient of the division is the constant

1

: the line to test is

y = 1

For a horizontal asymptote, the function must tend to a finite value: check the limit of

\frac{x + 2}{x + 4}

at both infinities

x

tends to +∞,

\frac{x + 2}{x + 4}

tends to

1

; as

x

tends to -∞,

\frac{x + 2}{x + 4}

tends to

1

The horizontal asymptote is

y = 1

; there is no slant asymptote

▶Asymptotes:

{x = -4, y = 1}

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