A function $f(x)$ defined as $f(x)=\left\{\begin{array}{ll}\frac{x^2-9}{x-3} & x \neq 3 \\ \text { undefined } & x=3\end{array}\right.$
Is there a removable discontinuity at $x=3$ ? Justify your answer.
Answer (1)
Simplify the function: f ( x ) = x − 3 x 2 − 9 = x + 3 for x = 3 .
Calculate the limit as x approaches 3: lim x → 3 f ( x ) = 6 .
Observe that f ( 3 ) is undefined, but the limit exists.
Conclude that there is a removable discontinuity at x = 3 , which can be removed by defining f ( 3 ) = 6 . The answer is yes, there is a removable discontinuity at x = 3 : T r u e .
Explanation
Problem Analysis We are given a function f ( x ) defined as
f ( x ) = { x − 3 x 2 − 9 undefined x = 3 x = 3
We need to determine if there is a removable discontinuity at x = 3 and justify our answer. A removable discontinuity occurs at a point x = a if the limit of the function as x approaches a exists, but the function is either not defined at x = a or the value of the function at x = a is not equal to the limit.
Simplifying the Function First, let's simplify the expression for f ( x ) when x = 3 :
f ( x ) = x − 3 x 2 − 9 = x − 3 ( x − 3 ) ( x + 3 )
Since x = 3 , we can cancel the ( x − 3 ) terms:
f ( x ) = x + 3 for x = 3 .
Calculating the Limit Now, let's find the limit of f ( x ) as x approaches 3:
lim x → 3 f ( x ) = lim x → 3 ( x + 3 ) = 3 + 3 = 6
The limit exists and is equal to 6.
Conclusion The function f ( x ) is undefined at x = 3 , but the limit as x approaches 3 is 6. Therefore, there is a removable discontinuity at x = 3 . We can remove the discontinuity by defining f ( 3 ) = 6 .
Justification The function f ( x ) has a removable discontinuity at x = 3 because the limit of f ( x ) as x approaches 3 exists (and is equal to 6), but f ( 3 ) is undefined.
Examples
Removable discontinuities are important in various fields, such as signal processing and image analysis. For example, when processing a signal, a removable discontinuity might represent a missing data point. By understanding the behavior of the function around the discontinuity, we can often 'fill in' the missing value by taking the limit, thus smoothing the signal and improving the quality of the processed data. This is similar to how image editing software can remove blemishes or imperfections by interpolating the surrounding pixel values.
