Problem 27 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 3} \frac{1}{(x-3)^{2}} $$

Step-by-Step Solution

Answer

The limit is infinite; \( \lim_{x \to 3} \frac{1}{(x-3)^2} = \infty \).

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{(x-3)^2} \). We want to find the limit of this function as \( x \) approaches 3.

Step 2: Create a Table of Values

Select values of \( x \) that approach 3 from both the left (e.g., 2.9, 2.99, 2.999) and the right (e.g., 3.1, 3.01, 3.001). Calculate \( f(x) \) for each of these values to observe the trend as \( x \) approaches 3.

Step 3: Calculate Values from the Table

Evaluate \( f(x) = \frac{1}{(x-3)^2} \):- For \( x = 2.9 \), \( f(x) = \frac{1}{(2.9-3)^2} = 100 \).- For \( x = 2.99 \), \( f(x) = \frac{1}{(2.99-3)^2} = 10000 \).- For \( x = 2.999 \), \( f(x) = \frac{1}{(2.999-3)^2} = 1000000 \).- For \( x = 3.1 \), \( f(x) = \frac{1}{(3.1-3)^2} = 100 \).- For \( x = 3.01 \), \( f(x) = \frac{1}{(3.01-3)^2} = 10000 \).- For \( x = 3.001 \), \( f(x) = \frac{1}{(3.001-3)^2} = 1000000 \).

Step 4: Analyze the Table

From the values in the table, as \( x \) approaches 3 from both the left and right, \( f(x) \) increases without bound, indicating that \( f(x) \to \infty \). This suggests that the limit does not exist in the traditional sense of converging to a finite number.

Key Concepts

Approaching Values

In calculus, the concept of 'approaching values' is pivotal when discussing limits. When we say that \( x \) approaches a certain value, like in our exercise where \( x \) approaches 3, we are interested in understanding the behavior of the function as it gets closer and closer to this value.

Approaching a value doesn't mean reaching it exactly. It involves checking what happens on either side of the number. This kind of exploration helps us understand the tendencies or trends in the function's behavior. For the function \( f(x) = \frac{1}{(x-3)^2} \), we observe what happens to the value of the function as \( x \) gets extremely close to 3 from both the left (e.g., 2.9, 2.99, 2.999) and the right (3.1, 3.01, 3.001).

As you work through limits, think about approaching values as zooming in closer and closer to a point, and always consider values from both sides. This approach helps you understand how the function behaves in the vicinity of that point.

Infinite Limits

Infinite limits occur when a function increases or decreases without bound as the input approaches a certain value. In the context of the provided function, infinite limits are at play. As \( x \) gets closer and closer to 3, whether from the left or the right, the function \( f(x) = \frac{1}{(x-3)^2} \) starts to shoot upwards and becomes very large.

This rapid increase indicates that the output of the function is heading towards infinity. Calculus refers to this as an infinite limit. What essentially happens here is that as \( x \) nears the critical point of 3, the denominator \((x-3)^2\) approaches zero, which makes the whole fraction become very large. Hence, instead of the function settling at a finite number, it heads off to infinity.
The notation used \( \lim_{x \to 3} f(x) = \infty \) signifies that as \( x \) nears 3, \( f(x) \) doesn't approach a fixed number, but rather increases without bound. This is a key concept when dealing with vertical asymptotes and points where the function 'blows up'.

Table of Values

A table of values is an excellent tool for investigating limits because it allows you to empirically observe how a function behaves as its input values approach a particular number. In our step-by-step solution, a table of values was used to tabulate results for \( f(x) \) by selecting \( x \)-values that target 3 from either direction.

By plugging these values into the function \( f(x) = \frac{1}{(x-3)^2} \), we could clearly observe a pattern. The calculated values (100, 10,000, 1,000,000) from both sides grew substantially larger as \( x \) got closer to 3. This tabular method is visually and numerically informative, making it easier to understand the concept of limits.

Tables help you see trends and validate results. When analyzing limits, crafting a table like this allows you to see that as the difference between \( x \) and 3 diminishes, the output of \( f(x) \) increases immensely. This growth pattern is a powerful visual indicator of an infinite limit.