Question

identify whether each value of x is a discontinuity of the function by typing asymptote, hole, or neither.\\(\frac{5x}{x^{3}+5x^{2}+6x}\\)\\(x = -3\\)\\(x = -2\\)\\(x = 0\\)\\(x = 2\\)\\(x = 3\\)\\(x = 5\\)

Explanation:

Step1: Factor denominator

Factor $x^3 + 5x^2 + 6x$:
$x(x^2 + 5x + 6) = x(x+2)(x+3)$
The function becomes $\frac{5x}{x(x+2)(x+3)}$

Step2: Simplify the function

Cancel common factor $x$ (for $x
eq 0$):
$\frac{5}{(x+2)(x+3)}$

Step3: Identify discontinuities

• Holes occur at canceled common factors: $x=0$
• Vertical asymptotes occur at remaining zeroes of denominator: $x=-2, x=-3$
• Values not making denominator zero are neither: $x=2,3,5$

Answer:

$x=-3$: asymptote
$x=-2$: asymptote
$x=0$: hole
$x=2$: neither
$x=3$: neither
$x=5$: neither