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