Problem 18 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{1}{x^{2}-1} $$

Step-by-Step Solution

Answer

The limit is -1 as \( x \to 0 \).

Step 1: Understand the problem

We need to find \( \lim _{x \rightarrow 0} \frac{1}{x^{2}-1} \). This involves investigating how the function behaves as \( x \) approaches 0.

Step 2: Identify undefined points

The expression \( \frac{1}{x^2 - 1} \) is undefined when \( x^2 - 1 = 0 \), which is when \( x = 1 \) and \( x = -1 \). However, the limit as \( x \to 0 \) does not approach these values.

Step 3: Approach limit from left and right

Consider how \( \frac{1}{x^2 - 1} \) behaves as \( x \) approaches 0 from the left (negative side) and right (positive side). Since \( x \to 0 \), \( x^2 - 1 \to -1 \), so the fraction approaches \( \frac{1}{-1} = -1 \).

Step 4: Verify with a table

Create a table with values approaching 0 from both negative and positive directions. For values such as \(-0.1, -0.01, 0.01, 0.1\), compute \( \frac{1}{x^2 - 1} \). For all these values, the result approaches -1.

Key Concepts

Approaching Values

When we talk about approaching values in calculus, we are concerned with how the function behaves as the input gets closer to a specific number. In this scenario, we are examining the limit of the function \( \frac{1}{x^2-1} \) as \( x \) approaches 0. This means observing the tendency of the function when the value of \( x \) nears 0 from both the negative and positive sides. The beauty of limits is that they allow us to predict the behavior of functions at points where they are not directly defined. Here, as \( x \to 0 \), the function approaches \(-1\) because \( x^2-1\) moves closer to \(-1\). Understanding this movement is key to grasping limits. When you approach a problem of this nature, visualize the graph or think about it: as \( x \) gets smaller towards zero, how does the expression adjust? This insight is foundational for verifying and understanding limits.

Undefined Points

In calculus, undefined points are spots where a function doesn't exist on the graph, which usually occurs when you divide by zero or encounter other problematic expressions. For the function \( \frac{1}{x^2-1} \), it's undefined where \( x^2-1=0 \). Simply put, if we solve \( x^2-1=0 \), we find that \( x=1 \) and \( x=-1 \) are undefined points. Although these points are problematic for the direct evaluation of the function, we are interested in examining the limit specifically as \( x \) approaches 0, which bypasses these undefined points. By understanding and identifying undefined points, you can focus on the segments of the function that actively describe its behavior around your point of interest.

Table of Values

A table of values can offer a clear visual insight into how a function behaves as a variable approaches a certain point. This is particularly helpful when dealing with limits. To build a table for the limit \( \lim_{x \to 0} \frac{1}{x^2-1} \):

• Choose values close to 0 from both sides, like -0.1, -0.01 (from the left), and 0.01, 0.1 (from the right).
• Calculate the function \( \frac{1}{x^2-1} \) for these values.
• Observe the results and look for a consistent pattern.

The calculations here show that the outputs of the function are hovering around \(-1\), suggesting that is the limit value. Using a table not only confirms calculations but helps visualize the behavior and approach of the function.

Limit Investigation

To fully comprehend the notion of limit investigation, it is important to analyze how values behave when closing in on a specific number. In the context of our problem, limit investigation involves understanding the behavior of \( \frac{1}{x^2 - 1} \) as \( x \to 0 \). To investigate this, you break it down:

• Determine the main characteristics of the function around \( x=0 \).
• Scrutinize the trend as it nears from both sides; \( x \to 0^+ \) yields a similar outcome to \( x \to 0^- \).

In this example, the result shows a consistent approaching value of \(-1\). Effective limit investigation empowers you to make predictions of behavior in mathematical functions, making this an essential skill in calculus. This methodical examination ensures deeper comprehension and provides the tools needed to tackle more complex functions in the future.