\[ f(x) = \left\{ \begin{array} $\dfrac{ 1 }{ x – 4 }\ where\ x \ne 4\ \\ 1 \hspace{0.3in} where\ x\ = 4 \end{array} \right. \]

The question aims to find why the **function f(x)** is **discontinuous** at the given **number a.**

The concept needed for this question includes **limits. Limit** is the approaching **value** of the **function** when the **input** of the **function** is also approaching some **value.** A **discontinuous function** is a **function** that is discontinuous at a **specific point** that has either a **left-hand limit not equal** to the **right-hand limit** or the function is **not defined** at that **point.**

## Expert Answer

The f(x) is given and it is **discontinuous** at **a=(4, y).** The **graph** of the **function** is shown below in Figure 1.

We can observe from the **graph** that the **function f(x)** has no defined value at **x=4.** We can use the definition of the **discontinuous function** to explain why the **function f(x)** is **discontinuous** at **x=4.**

According to the definition, a function is **discontinuous** if its **left-hand** and **right-hand limits** are **not equal.** The **right-hand limit** of the function is given as:

\[ \lim_{x \rightarrow a^+} f(x) = f(a) \]

\[ \lim_{x \rightarrow a^+} f(x) = + \infty \]

The **right-hand limit** is approaching **positive infinity.** The **left-hand limit** is given as:

\[ \lim_{x \rightarrow a^-} f(x) = f(a) \]

\[ \lim_{x \rightarrow a^-} f(x) = – \infty \]

The **left-hand limit** is approaching **negative infinity.** Here **a=4,** the input of the function approaches **a**, and **limits** are approaching **infinities** at **x=4.**

Thus, we can conclude that the **function f(x)** is **discontinuous** at **a=4** according to the definition of the discontinuous function.

## Numerical Result

The given **function f(x)** is a **discontinuous function** as its **left-hand limit** is** not equal** to the **right-hand limit** which is a requirement according to its definition.

## Example

Explain the given **function f(x)** is **discontinuous** at **x=2** and sketch its graph.

\[ f(x) = \dfrac{ 1 }{ x\ -\ 2 }\ where\ x \ne 2 \]

The **graph** of the **function** is shown below in Figure 2.

The **right-hand limit** of the function is given as:

\[ \lim_{x \rightarrow a^+} f(x) = f(a) \]

\[ \lim_{x \rightarrow a^+} f(x) = + \infty \]

The **right-hand limit** is approaching **positive infinity.** The **left-hand limit** is given as:

\[ \lim_{x \rightarrow a^-} f(x) = f(a) \]

\[ \lim_{x \rightarrow a^-} f(x) = – \infty \]

The **left-hand limit** is approaching **negative infinity.** Here **a=2,** the input of the function approaches **a,** and **limits** are approaching **infinities** at **x=2.**

Thus we can conclude that the **function f(x)** is **discontinuous** at **a=2,** as its **left-hand limit** is **not equal** to its **right-hand limit.** Hence satisfying the **definition** of the **discontinuous function.**