Question

consider the table representing a rational function.
x | -0.1 | -0.01 | -0.001 | 0 | 0.001 | 0.01 | 0.1 | 2.9 | 3 | 3.1 | 4.9 | 4.99 | 5 | 5.001 | 5.01 | 5.1
f(x) | 1.96 | 19.96 | 199.96 | undefined | -200.04 | -20.04 | -2.04 | -0.16 | undefined | -0.17 | -2.04 | -20.04 | undefined | 199.96 | 19.96 | 1.96
which statement describes the graph of the function?
○ the function has holes when x = 0, x = 3, and x = 5.
○ the function has vertical asymptotes when x = 0, x = 3, and x = 5.
○ the function has holes when x = 0 and x = 3, and a vertical asymptote when x = 5.
○ the function has a hole when x = 3, and vertical asymptotes when x = 0 and x = 5.

Explanation:

1. For a rational function, a hole occurs at an x-value where the function is undefined, but the left-hand and right-hand limits as x approaches that value are equal (finite). A vertical asymptote occurs at an x-value where the function is undefined, and the left-hand and right-hand limits approach positive or negative infinity (and the limits from both sides are not equal finite values).
2. At $x=0$: As $x$ approaches 0 from the left, $f(x)$ approaches $+\infty$; as $x$ approaches 0 from the right, $f(x)$ approaches $-\infty$. This matches a vertical asymptote.
3. At $x=3$: As $x$ approaches 3 from the left, $f(x)\approx -0.16$; as $x$ approaches 3 from the right, $f(x)\approx -0.17$. These are finite, nearly equal values, so this is a hole.
4. At $x=5$: As $x$ approaches 5 from the left, $f(x)$ approaches $-\infty$; as $x$ approaches 5 from the right, $f(x)$ approaches $+\infty$. This matches a vertical asymptote.

Answer:

The function has a hole when x = 3, and vertical asymptotes when x = 0 and x = 5.