Problem 28 - Calculus Early Transcendentals

Question

Determining limits analytically Determine the following limits or state that they do not exist. $$\lim _{z \rightarrow 4} \frac{z-5}{\left(z^{2}-10 z+24\right)^{2}}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as z approaches 4 is -1.

Step 1: Direct substitution

Try direct substitution by replacing z with 4: $$\lim _{z \rightarrow 4} \frac{z-5}{\left(z^{2}-10 z+24\right)^{2}} = \frac{4-5}{\left(4^{2}-10(4)+24\right)^{2}}$$

Step 2: Simplification

Simplify the expression: $$\frac{-1}{\left(16-40+24\right)^{2}} = \frac{-1}{\left(0\right)^{2}}$$

Step 3: Determine the limit

Since the simplified expression evaluates to a well-defined value without any indeterminacy, we can conclude that$$\lim _{z \rightarrow 4} \frac{z-5}{\left(z^{2}-10 z+24\right)^{2}}=\frac{-1}{0^2}=-1$$ Therefore, the limit of the function as z approaches 4 is -1.

Key Concepts

Limits in Calculus

In calculus, the concept of a 'limit' is one of the fundamental building blocks. It essentially asks the question: As we approach a certain point, what value does a function get close to?

For example, when we look at a function graphically, there can be points where it's not immediately clear what the function's behavior is at that point. This uncertain behavior could be due to a hole in the graph, a vertical asymptote, or the point itself not being included in the domain of the function. Here's where limits come in handy—they help us describe the behavior of functions at points that are not immediately clear.

Understanding the concept of limits is essential for further study in calculus, such as when learning about continuity, derivatives, and integrals. All of these concepts rely on the understanding of how functions behave as they get close to certain values, even if those values are not within the domain of the function.

Direct Substitution Method

The direct substitution method is one of the first techniques you should try when evaluating limits. This approach is exactly as it sounds: You directly substitute the value that the variable is approaching into the function, if possible.

If the function is continuous at the point you're considering, direct substitution will give you the exact value of the limit. But, beware of situations where substituting the value leads to undefined expressions, such as dividing by zero or getting the indeterminate form 0/0. In those cases, further analysis is required and direct substitution isn’t sufficient.

In the provided exercise, the direct substitution of z with 4 initially suggested an undefined situation. However, a closer look revealed that the function indeed simplifies to a defined value, allowing the use of the direct substitution method for a seamless limit evaluation.

Limit Evaluation

Evaluating limits is a critical skill in calculus and involves determining what value a function approaches as the input approaches a certain value. The direct substitution method is often the first step in limit evaluation. However, when it results in an undetermined form, other techniques like factoring, rationalizing, or applying L'Hopital's rule may be necessary.

In the given problem where the limit as z approaches 4 was sought, we saw that after substituting 4 into the function, the denominator initially seemed to pose an issue as it resulted in 0. But a closer look showed that it actually simplifies to a division by zero situation, which signals that the function is undefined at the exact value we are considering. Yet, the limit as z approaches 4 can still be evaluated through direct substitution, leading to a meaningful result.

It's important to recognize that the process of evaluating limits is about understanding behavior near a point, not necessarily at that point. In the context of this exercise, although the function is not defined at z = 4, the procedure revealed the limit's value effectively.