Answer
Vertical: $x = -3$
Horizontal: $y = 4$
Work Step by Step
To find the horizontal asymptote, we need to find:
$\lim\limits_{x \to \infty} \frac{5 + 4x}{x+3}$ and $\lim\limits_{x \to -\infty} \frac{5 + 4x}{x+3}$
First we divide both numerator and denominator by $x$.
$\lim\limits_{x \to \infty} \frac{\frac{5+4x}{x}}{\frac{x+3}{x}} =$
$\lim\limits_{x \to \infty} \frac{\frac{5}{x}+4}{\frac{3}{x}+1} = \frac{0+4}{1+0} = 4$
Now with $-\infty$.
$\lim\limits_{x \to -\infty} \frac{\frac{5+4x}{x}}{\frac{x+3}{x}} =$
$\lim\limits_{x \to -\infty} \frac{\frac{5}{x}+4}{\frac{3}{x}+1} = \frac{0+4}{1+0} = 4$
So with both answers we know that the horizontal asymptote is $y = 4$.
To find the vertical asymptote we take the denominator $x+3$ and compare it to zero.
$x + 3 = 0$
$x = -3$
So the vertical asymptote is $x = -3$ and the horizontal asymptote is $y = 4$.
