Problem 35 - Contemporary Precalculus

Question

Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow 4} \frac{-6}{(x-4)^{2}}$$

Step-by-Step Solution

Answer

Answer: The given function has no limit as x approaches 4, and the function value at x = 4 is also undefined. Therefore, we cannot compare them.

Step 1: Check the limit as x approaches 4

Let's rewrite the given limit: $$\lim _{x \rightarrow 4} \frac{-6}{(x-4)^{2}}$$ Now let's check if there is an indeterminate form, which means we should first plug in x = 4 and see if we have something like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Plug in x = 4: $$\lim _{x \rightarrow 4} \frac{-6}{(4-4)^{2}} = \frac{-6}{0^{2}} = \frac{-6}{0}$$ As we can see, this is a case where the denominator becomes zero, and therefore the limit is undefined. Now, let's check the value of the function at x=4, if it exists.

Step 2: Verify if the function value exists at x = 4

Since there's no explicit formula for the given function f, we only have its expression: $$f(x) = \frac{-6}{(x-4)^{2}}$$ Plug in x = 4: $$f(4) = \frac{-6}{(4-4)^{2}} = \frac{-6}{0^{2}} = \frac{-6}{0}$$ The function value is also undefined when x = 4.

Step 3: State the conclusion about the limit of f(x) and compare with f(c)

From Step 1, we found out that the limit $$\lim _{x \rightarrow 4} \frac{-6}{(x-4)^{2}}$$ is undefined since the denominator becomes zero when x approaches 4. From Step 2, we found out that the function value is also undefined when x = 4. Since the limit of the function as x approaches 4 does not exist and likewise the function value is undefined at x = 4, we conclude that the given function has no limit as x approaches 4 and cannot compare it to the function value at x=4 (f(c)), as they both are undefined.

Key Concepts

Undefined Limits

When you study limits, sometimes you'll encounter a situation where substituting a value results in a zero in the denominator. This means the expression becomes undefined, because dividing by zero isn't possible in mathematics. In our example, plugging in 4 into the function \( f(x) = \frac{-6}{(x-4)^2} \) leads to the denominator becoming \( 0^2 \), leaving us with \( \frac{-6}{0} \). For this reason, the limit is deemed undefined.

Undefined limits occur in several scenarios, such as:

• When a function blows up to infinity as you approach the point of interest.
• When there is a sudden jump in the function values.
• When the function oscillates erratically near that point.

It's important to note that a limit being undefined doesn't always mean there's no information to be gained. Sometimes, recognizing an undefined limit can give hints about the behavior of a function as it nears certain points.

Evaluating Limits

Evaluating a limit typically involves several strategies. You usually start by substituting the approaching value into the function. If this results in an undefined expression, which is often the case with zeros in the denominator, other techniques must be employed. These include simplifying the expression, using L'Hôpital's Rule, or graphically analyzing the function.

In our exercise, evaluating \( \lim_{x \rightarrow 4} \frac{-6}{(x-4)^2} \) resulted directly in a denominator of zero, highlighting an undefined limit. Attempting to simplify or refactor might not always help, especially when asymptotes indicate the function's approach towards infinity or negative infinity as in this example.

To effectively evaluate limits, understanding both algebraic manipulation and recognizing special forms, like indeterminate forms, is essential. If none of these provide insights, moving towards graphical approaches or numerical estimation might be necessary to gather more information.

Graphical Approach to Limits

A graphical approach to limits involves visualizing the function's behavior as it approaches a specific value. By plotting \( y = \frac{-6}{(x-4)^2} \), you observe that as \( x \) gets closer to 4, the values of the function increase without bound. This visually confirms why the limit is undefined at \( x = 4 \), due to vertical asymptote at this point.

Graphical methods are beneficial for:

• Identifying if a function approaches infinity (positive or negative) or some finite value.
• Understanding the presence of discontinuities or jumps in the graph.
• Verifying algebraic limit calculations through visual insights.

When you use a graph, it's easier to recognize patterns in how the function behaves around the point of interest. However, always remember that the graphical method relies heavily on the resolution of the graph which sometimes might not show very subtle details present near the limit point.