asymptotes of (x-9)/(sqrt(4x^2+3x+2))

1) Let's find the asymptotes of

\frac{x - 9}{\sqrt{4 x^{2} + 3 x + 2}}

, for real

x

The denominator is a square root: the domain is given by

4 x^{2} + 3 x + 2 > 0

⇥1.1) Let's solve in

ℝ

4 x^{2} + 3 x + 2 > 0

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

⇥

Δ = 3^{2} - 4 \cdot 4 \cdot 2

⇥ℹ

3^{2}

9

‖

4 \cdot 4

16

⇥

Δ = 9 - 16 \cdot 2

⇥

Δ = 9 - 32

⇥

Δ = -23

⇥Since a =

4

> 0, the solution is:

ℝ

⇥Solution obtained:

x \in ℝ

The radicand is strictly positive on ℝ: the function is defined and continuous on ℝ, with no vertical asymptote

At infinity, the dominant terms are

x

in the numerator and

\sqrt{4 x^{2}}

in the denominator. Remember:

\sqrt{x^{2}}

abs (x)

x

tends to

-∞

\frac{x}{abs (x)}

-1

: the function tends to

- \frac{1}{2}

-∞

, the limit is finite:

y = - \frac{1}{2}

is a horizontal asymptote

x

tends to

+∞

\frac{x}{abs (x)}

1

: the function tends to

\frac{1}{2}

+∞

, the limit is finite:

y = \frac{1}{2}

is a horizontal asymptote

The limits at both infinities are finite: there is no slant asymptote

▶Asymptotes:

{y = - \frac{1}{2}, y = \frac{1}{2}}

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