AP Calculus AB: 1.10 Exploring Types of 3 Discontinuities - Exam Style questions with Answer- FRQ

The function $f(x)$ is defined piecewise as shown in the graph below:

Using the graph above, answer the following:

(a) Identify the type of discontinuity at each of the following points:

$x = -2$
$x = 0$
$x = 2$
$x = 4$

(b) Justify your answers using the definition of limits and continuity.

▶️ Answer/Explanation

Explanation of Discontinuities

1. Jump Discontinuity

At $x = -2$:

$
\lim_{x \to -2^-} f(x) = 1, \quad \lim_{x \to -2^+} f(x) = 1, \quad f(-2) = -2
$

So, $\lim_{x \to -2} f(x) \neq f(-2) \Rightarrow$ Jump discontinuity

Also at $x = 4$:

$
\lim_{x \to 4^-} f(x) = \sin(4) \approx -0.7568, \quad f(4) = 0
$

2. Removable Discontinuity

At $x = 2$:

$
\lim_{x \to 2} f(x) = \sin(2) \approx 0.9093, \quad f(2) = 1.5
$

Function is defined but not equal to the limit ⇒ Removable discontinuity

3. Infinite Discontinuity

At $x = 0$: Function is undefined due to a break (division-type behavior or asymptote)

$
\text{Simulated vertical asymptote (annotated in green).}
$

Describe the discontinuities of the both function below.

▶️ Answer/Explanation

Figure 6:

There is a jump discontinuity at $x = -1$ and an infinite discontinuity at $x = 2$

Figure 7:

There are jump discontinuities at $x = -2$ and $x = 4$. There is a removable discontinuity at $x = 2$. There is an infinite discontinuity at $x = 0$.

AP Calculus AB: 1.10 Exploring Types of 3 Discontinuities – Exam Style questions with Answer- FRQ