Problem 14
Question
In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to -2} \dfrac{x+2}{x^2+5x+6}$$
Step-by-Step Solution
Verified Answer
The limit of the function \(f(x) = \dfrac{x+2}{x^2+5x+6}\) as \(x \to -2\) can be found by substituting values close to -2 into the function and observing the corresponding outputs. A graph of the function will also help to confirm the result. But, because the actual computations and graphs are not provided here, the exact value of the limit is not provided.
1Step 1: Understand the Function
The function we are given is \(f(x) = \dfrac{x+2}{x^2+5x+6}\). We are required to find the limit of this function as x approaches -2.
2Step 2: Create a Table of Values
For the table of values, choose values of x that are close to -2 from both sides. For example, -2.1, -2.01, -1.99, and -1.9. Substitute these values into the function and record the corresponding outputs.
3Step 3: Observe the Behavior of the Outputs
As the values of x get closer and closer to -2 from both sides, do the outputs approach a certain value? This approached value is the limit of the function as x approaches -2, if it exists.
4Step 4: Graph the Function
Using a graphing utility, plot the graph of \(y = f(x)\), where \(f(x) = \dfrac{x+2}{x^2+5x+6}\). Observe the behavior of the function as x approaches -2 from the graph. Does it approach the same value as reported in the table of values?
Key Concepts
Limits of FunctionsCreating a Table of ValuesGraphing Utility AnalysisApproaching Limits Conceptually
Limits of Functions
Understanding the concept of a limit is fundamental when studying calculus. The limit of a function describes the behavior of the function as the input values approach a certain point or value. For example, when we analyze \(\lim_{x \to -2} \dfrac{x+2}{x^2+5x+6}\), we want to determine what the output values of the function become as the input values get very close to -2.
Imagine standing at a point on a path and walking towards a signpost; as you get closer, you observe more details about the signpost. The limit describes what value, if any, the function approaches, not necessarily the function's value exactly at that point (the signpost). Sometimes the function may not be defined at that point, but we are still interested in its behavior as we approach it.
Imagine standing at a point on a path and walking towards a signpost; as you get closer, you observe more details about the signpost. The limit describes what value, if any, the function approaches, not necessarily the function's value exactly at that point (the signpost). Sometimes the function may not be defined at that point, but we are still interested in its behavior as we approach it.
Creating a Table of Values
A practical approach to estimating limits numerically involves creating a table of values. By choosing input values that are progressively closer to the point of interest from both sides, we can observe the function's output values. For instance, if we're approaching -2, we might select values like -2.1, -2.01, and even -2.001 to see how the function behaves as we get closer from the left and similarly, -1.9, -1.99, and -1.999 from the right.
To be effective, ensure that your input values are spaced consistently, and note down the function outputs. The pattern in these values can indicate the limit. If the outputs stabilize or converge toward a specific number, that is the limit you're estimating. Remember, a table is a tool: it's up to your analysis to interpret the results correctly.
To be effective, ensure that your input values are spaced consistently, and note down the function outputs. The pattern in these values can indicate the limit. If the outputs stabilize or converge toward a specific number, that is the limit you're estimating. Remember, a table is a tool: it's up to your analysis to interpret the results correctly.
Graphing Utility Analysis
Graphing utilities are powerful tools for visualizing and understanding the behavior of functions. After creating a table of values, graphing the function can confirm the limit we've estimated numerically.
To graph our given function \(f(x) = \dfrac{x+2}{x^2+5x+6}\), use a graphing utility that plots points and draws a curve representing the function. By zooming in on the graph around the value x = -2, we can watch the behavior of the curve. If the function's graph approaches a specific y-value as it gets closer to x = -2, this visual cue provides confirmation for our numerical estimate. Graphing utilities can be particularly helpful for spotting patterns that aren't obvious from the table alone and for verifying the behavior in cases where the limit is less intuitive.
To graph our given function \(f(x) = \dfrac{x+2}{x^2+5x+6}\), use a graphing utility that plots points and draws a curve representing the function. By zooming in on the graph around the value x = -2, we can watch the behavior of the curve. If the function's graph approaches a specific y-value as it gets closer to x = -2, this visual cue provides confirmation for our numerical estimate. Graphing utilities can be particularly helpful for spotting patterns that aren't obvious from the table alone and for verifying the behavior in cases where the limit is less intuitive.
Approaching Limits Conceptually
Approaching a limit conceptually means understanding the 'why' behind the behavior of a function near a point, not just the 'what'. It involves thinking about how the function’s formula will behave as the input gets indefinitely closer to a target value and what that implies about the output.
With our example function \(f(x) = \dfrac{x+2}{x^2+5x+6}\), consider why the outputs look the way they do as we near -2. Are there factors in the function that become negligible or dominant? Understanding the factors of the denominator, in this case, helps us to see that, as x approaches -2, the function is not approaching infinity or zero, but rather, is there a cancellation or a looming undefined value at that point? Conceptual analysis complements numerical and graphical methods, giving a more complete picture of the function's behavior near a limit.
With our example function \(f(x) = \dfrac{x+2}{x^2+5x+6}\), consider why the outputs look the way they do as we near -2. Are there factors in the function that become negligible or dominant? Understanding the factors of the denominator, in this case, helps us to see that, as x approaches -2, the function is not approaching infinity or zero, but rather, is there a cancellation or a looming undefined value at that point? Conceptual analysis complements numerical and graphical methods, giving a more complete picture of the function's behavior near a limit.
Other exercises in this chapter
Problem 14
In Exercises 9-16, use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result. \
View solution Problem 14
In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to -3} \dfrac{2x^2+5x-3}{x+3}$$
View solution Problem 15
In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x
View solution Problem 15
In Exercises 9-16, use the limit process to find the slope of the graph of the function at the specified point. Use a graphing utility to confirm your result. \
View solution