Problem 13 - Contemporary Precalculus

Question

Explain why the function is not continuous at the given number. $$f(x)=1 /(x-3)^{3} \quad \text { at } x=3$$

Step-by-Step Solution

Answer

Answer: The function $$f(x) = \frac{1}{(x-3)^3}$$ is not continuous at \(x=3\) because it is not defined at that point (division by zero) and its limit does not exist as \(x\) approaches 3.

Step 1: Condition 1: Function value at x=3

Let's try to find the value of the function when \(x = 3\). $$f(3) = \frac{1}{(3-3)^3}$$ We cannot compute this value because \((3-3)^3 = 0^3 = 0\), and division by zero is undefined. Since the function is not defined at \(x=3\), this condition fails. Thus, the function is not continuous at \(x=3\).

Step 2: Condition 2: Limit existence at x=3

For the sake of completeness, let's check if the limit of the function exists as \(x\) approaches 3. $$\lim_{x\to3} f(x) = \lim_{x\to3} \frac{1}{(x-3)^3}$$ As \(x\) approaches 3, the denominator \((x-3)^3\) approaches 0, causing the overall expression to approach infinity. Therefore, the limit does not exist as \(x\) approaches 3.

Step 3: Conclusion

Since the function is not defined at \(x=3\) and its limit does not exist at that point, we can conclude that the function $$f(x) = \frac{1}{(x-3)^3}$$ is not continuous at \(x=3\).

Key Concepts

Discontinuity

In mathematics, a function is said to be continuous at a certain point if there is no interruption in its graph at that point. In simpler terms, you can draw the function without lifting your pencil off the paper. However, when there's a disruption or a gap in the function at a specific point, we call this a discontinuity.
In the given problem, the function \(f(x)=\frac{1}{(x-3)^3}\) is not continuous at \(x=3\). This is because the function is not defined at this point. When the function "jumps" or has holes in it, it signals a discontinuity.

To test for continuity, we must check these conditions at \(x=3\):

❋ The function must be defined at \(x=3\). ❋ The limit as \(x\) approaches 3 must exist. ❋ The function value and the limit must be equal.
Since these conditions are not met, the function is discontinuous at \(x=3\). This is a key reason why understanding discontinuity is crucial to analyzing functions.

Limit

The concept of a limit is fundamental in calculus. A limit refers to the value that a function approaches as the input approaches a certain point. Even if the function is not defined at that point, the limit helps us understand the behavior of the function near that point.
For the function \(f(x) = \frac{1}{(x-3)^3}\), we are asked about its limit at \(x=3\). As \(x\) gets closer and closer to 3, the expression \((x-3)^3\) approaches 0. This results in the fraction approaching infinity because we're dividing by an increasingly small number.
Here's the important part:

• The limit does not exist because the function value shoots off to infinity, indicating an infinite discontinuity.
• An infinite limit results when a function becomes unbounded near a point it's undefined.

Understanding limits helps us deal with situations where the function value might be undefined yet give us important information about function behavior.

Division by Zero

Division by zero is a mathematical operation that is undefined. This happens when you try to divide a number by zero, which is not possible under real number arithmetic.
In our function \(f(3) = \frac{1}{(3-3)^3}\), substituting \(x=3\) directly into the function leads to dividing 1 by zero, which is 0. This makes the function undefined at \(x=3\). Therefore, division by zero at this point results in the function being discontinuous.

• This point of discontinuity occurs because you can't assign a real number to division by zero.
• Mathematicians consider the function to "break down" at this point.

The idea of division by zero is fundamental to understanding why certain calculations might not be possible or lead to undefined results.

Undefined Function

The term "undefined function" sounds complex, but it essentially means that for some values of \(x\), we cannot compute \(f(x)\). This often occurs where there is division by zero.
Take the function \(f(x) = \frac{1}{(x-3)^3}\). If you plug in \(x=3\), you'll end up with an undefined value because the denominator becomes zero.
Understanding undefined functions involves:

• Realizing that for some points (like \(x=3\) here), no function value exists.
• Knowing that this absence of a function value implies a discontinuity.

Notably, when a function is undefined at a particular point, it suggests there's more than just a "hole" in the graph, but a lack of any calculable output at that input. This has real implications in calculus and mathematical analysis.