Problem 42 - Precalculus with Limits

Question

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \dfrac{\sqrt{x+5}-4}{x-2}, \quad \lim_{x \to 2} f(x)$$

Step-by-Step Solution

Answer

The conclusion will be dependent on the behavior of the graph as \(x\) approaches 2. If the function approaches a certain value, state that value as the limit. If it does not, explain that the limit does not exist due to the behavior of the function.

Step 1: Graphing the Function

Utilize a graphing utility to plot the function \(f(x) = \frac{\sqrt{x+5}-4}{x-2}\). Pay close attention to the behavior of the function near \(x = 2\).

Step 2: Assessing the Limit

Look at the graph and notice how the function behaves around \(x = 2\). If the function seems to be approaching a certain value, then the limit exists. If the function seems to be diverging or oscillating wildly, then the limit does not exist.

Step 3: Evaluating the Limit

Try to find the limit numerically using the graph. If the function appears to converge to a specific value, try to approximate it by looking at the y-values as \(x\) approaches 2 from either side. If the function does not appear to converge, state that the limit does not exist and provide a reasoning based on the behavior of the graph.

Key Concepts

Graphing Utilities

Graphing utilities are valuable tools that help us visualize mathematical functions and their behaviors. These can include software like graphing calculators, online graph plotters, or computer algebra systems. By plotting the function, we can better understand how it behaves at certain points of interest.
For the provided function \[f(x) = \frac{\sqrt{x+5} - 4}{x-2}\],using a graphing utility allows us to clearly see the behavior around the point \(x = 2\).
Here's what to do:

• Enter the function into the graphing tool.
• Observe the graph, specifically near \(x = 2\).
• Notice any holes, jumps, or asymptotes in the graph.

These observations will help determine whether the function has a limit at a specific point.

Limit Existence

The concept of limit existence helps us understand whether a function approaches a particular value as the input approaches some number. For the function given \[\lim_{x \to 2} \frac{\sqrt{x+5} - 4}{x-2}\],we need to ascertain what value, if any, the function approaches as \(x\) gets closer to 2.
When checking a graph:

• If the graph approaches the same value from both sides of the point in question, the limit is likely to exist.
• If the graph shows a hole or sudden jump, it might suggest a more complex behavior at that point that may prevent the limit from existing.

Specifically, you might find that this function has a removable discontinuity or behaves differently approaching from the left and right, thus indicating the limit's nature.

Numerical Approximation

When analyzing limits numerically, especially using graphical methods, numerical approximation takes center stage. This process involves observing the y-values that the function outputs as it gets closer to the desired x-value, in this case, \(x = 2\).
To perform a numerical approximation:

• Pick values of \(x\) that get incrementally closer to 2 from both the left (say, 1.9, 1.95) and right (e.g., 2.1, 2.05).
• Note the corresponding y-values.
• If these y-values seem to stabilize around a particular number, that suggests a numerical approximation of the limit.

This approach can be especially useful if visualization is unclear or if the graph shows a potential limit without exact resolution.