Problem 25
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. $$\lim_{x \to 1} \dfrac{\textrm{ln}(2x-1)}{x-1}$$
Step-by-Step Solution
Verified Answer
Based on the table of values created and the graph of the function, the estimated limit as x tends to 1 is around 2.
1Step 1: Create a table of values
Begin by creating the table with values of 'x' approaching 1 from both sides. For 'x' values, you can use 0.9, 0.99, 0.999, 1.001, 1.01, and 1.1. Compute the corresponding 'y' values using the function \(\frac{\ln(2x-1)}{x-1}\).
2Step 2: Estimate the limit numerically
Now, look at the 'y' values in the table. The limit is the number to which the 'y' value is getting closer as 'x' approaches 1 from both sides.
3Step 3: Graphical confirmation
Graph the function \(\frac{\ln(2x-1)}{x-1}\) using a graphing utility. Observe the behavior of the curve as 'x' approaches 1. The y-value it seems to be converging on is the limit.
Key Concepts
Limits of FunctionsNumerical Estimation in CalculusGraphing Utility in Calculus
Limits of Functions
Understanding the concept of limits is fundamental in calculus, as it helps us to describe the behavior of functions as they approach a specific point. A limit can be thought of as the value that a function 'approaches' as the input (or 'x') gets closer and closer to some number. It's important to note that the limit does not always equal the function's value at that point—if the function even has a value there. For example, the limit of \( \frac{\mathrm{ln}(2x-1)}{x-1} \) as x approaches 1 is not immediately clear because there's an apparent discontinuity at x=1.
When evaluating the limit, we essentially want to know what happens to \( \frac{\mathrm{ln}(2x-1)}{x-1} \) as x gets infinitely close to 1. This can be a bit like investigating a detective case, where clues are hidden in the function's behavior around that mysterious x value. 'Estimating limits numerically is akin to gathering evidence by observing the function's values at points increasingly near our point of interest—like detectives looking for patterns to solve a case.
Graphing can further help us visualize these patterns, by allowing us to see where the function heads as x approaches that crucial point.
When evaluating the limit, we essentially want to know what happens to \( \frac{\mathrm{ln}(2x-1)}{x-1} \) as x gets infinitely close to 1. This can be a bit like investigating a detective case, where clues are hidden in the function's behavior around that mysterious x value. 'Estimating limits numerically is akin to gathering evidence by observing the function's values at points increasingly near our point of interest—like detectives looking for patterns to solve a case.
Graphing can further help us visualize these patterns, by allowing us to see where the function heads as x approaches that crucial point.
Numerical Estimation in Calculus
Numerical estimation in calculus is akin to using a microscope to get a close-up view of a function's behavior as we near a certain point. In the case of our limit \( \lim_{x \to 1} \frac{\textrm{ln}(2x-1)}{x-1} \), we strategically choose x-values that are closer and closer to 1, from both sides of it. This narrowing approach helps students see the 'trend' in the y-values.
It’s like watching a runner approaching the finish line—while we can't tell their exact moment of crossing, we can estimate their position as they get closer. By analyzing a table of values, with each successive x-value getting infinitesimally closer to our point of convergence, we gather our numerical evidence. Even without a calculator, students can see the pattern emerge—providing them with the conceptual understanding that the value they're looking for is where the y-values are heading, not necessarily where they are at any step. This hands-on approach brings the abstract concept of limits into the realm of tangible numbers and patterns.
It’s like watching a runner approaching the finish line—while we can't tell their exact moment of crossing, we can estimate their position as they get closer. By analyzing a table of values, with each successive x-value getting infinitesimally closer to our point of convergence, we gather our numerical evidence. Even without a calculator, students can see the pattern emerge—providing them with the conceptual understanding that the value they're looking for is where the y-values are heading, not necessarily where they are at any step. This hands-on approach brings the abstract concept of limits into the realm of tangible numbers and patterns.
Graphing Utility in Calculus
A graphing utility acts as one of our most powerful tools in calculus. After we've done the detective work of estimating limits numerically, the graphing utility serves to visually confirm our findings. It's like taking the numerical data and transforming it into a striking photograph that reveals the full landscape of our function's behavior near a point.
In the exercise where we're considering \( \frac{\mathrm{ln}(2x-1)}{x-1} \) as x approaches 1, plotting this function gives us a visual aid to see if the y-values are indeed approaching a particular number. Just as an architect drafts a plan to visualize a structure before building, using a graph allows us to forecast where the function is headed—providing reassurance that our numerical estimation is on track.
Moreover, for visual learners, this method of graphical confirmation is invaluable. It not only enhances their understanding but also boosts their confidence in using mathematics to explain complex phenomena. By integrating graphing utilities into their toolset, students equip themselves with a dynamic method of investigation, enabling them to conquer even the most daunting calculus challenges.
In the exercise where we're considering \( \frac{\mathrm{ln}(2x-1)}{x-1} \) as x approaches 1, plotting this function gives us a visual aid to see if the y-values are indeed approaching a particular number. Just as an architect drafts a plan to visualize a structure before building, using a graph allows us to forecast where the function is headed—providing reassurance that our numerical estimation is on track.
Moreover, for visual learners, this method of graphical confirmation is invaluable. It not only enhances their understanding but also boosts their confidence in using mathematics to explain complex phenomena. By integrating graphing utilities into their toolset, students equip themselves with a dynamic method of investigation, enabling them to conquer even the most daunting calculus challenges.
Other exercises in this chapter
Problem 25
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 25
In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 0} \dfrac{\sqrt{2x+1} - 1}{x}$$
View solution Problem 26
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 26
In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 9} \dfrac{3 - \sqrt{x}}{x-9}$$
View solution