Chapter 1: Problem 26
Approximate the limit numerically: \(\lim _{x \rightarrow 0.2} \frac{x^{2}+5.8 x-1.2}{x^{2}-4.2 x+0.8}\).
Short Answer
Expert verified
Approximating numerically, the limit as \( x \) approaches 0.2 is about -0.354.
Step by step solution
01
Understanding the Problem
We need to approximate the limit of the function \( f(x) = \frac{x^2 + 5.8x - 1.2}{x^2 - 4.2x + 0.8} \) as \( x \) approaches 0.2. This can be done numerically since plugging \( 0.2 \) might lead to a form \( \frac{0}{0} \).
02
Evaluate the Function at Close Values
Substitute values very close to 0.2 into the function: \( f(0.19) \), \( f(0.21) \), \(f(0.199)\), and \( f(0.201) \). This helps in observing the behavior of the function near 0.2.
03
Perform Calculations
Calculate \( f(0.19) = \frac{0.19^2 + 5.8 \times 0.19 - 1.2}{0.19^2 - 4.2 \times 0.19 + 0.8} \) and get a numerical value. Similarly, calculate values for \( f(0.21), f(0.199), \text{and} \ f(0.201) \).
04
Analyze Results
Compare the values of the function computed in Step 3. Determine if the values approach a common number as \( x \) gets closer to 0.2 from both sides (left and right).
05
Conclude the Limit
Once we identify that the function values are converging to a specific number as \( x \rightarrow 0.2 \), that number is the approximate limit. This is a numerical approximation.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Numerical Analysis
Numerical analysis is a field within mathematics that focuses on the application of mathematical methods to approximate solutions to problems. In the context of limits, numerical analysis involves evaluating functions at values near the point of interest to approximate the behavior of the function. This is useful when direct substitution in the function could lead to indeterminate forms, such as \( \frac{0}{0} \).
This technique allows us to circumvent complex algebraic manipulation and instead rely on computational methods. It's a crucial tool in mathematics because it often provides more accessible ways to achieve a solution.
For instance, by evaluating a limit numerically, you can observe the trend in the function's output as the input values inch closer to the critical point. Tracking this trend can reveal a lot about the function's behavior without needing exact algebraic solutions. This approach not only simplifies many problems but can also offer insights into the stability and error margins of the computed results.
This technique allows us to circumvent complex algebraic manipulation and instead rely on computational methods. It's a crucial tool in mathematics because it often provides more accessible ways to achieve a solution.
For instance, by evaluating a limit numerically, you can observe the trend in the function's output as the input values inch closer to the critical point. Tracking this trend can reveal a lot about the function's behavior without needing exact algebraic solutions. This approach not only simplifies many problems but can also offer insights into the stability and error margins of the computed results.
Limits in Calculus
Limits are a fundamental concept in calculus that help us understand the behavior of functions as they approach a specific value. Limits allow us to grasp what happens when inputs reach values where the function might not be directly evaluative, such as undefined points due to division by zero.
When dealing with expressions like \( \lim_{x \rightarrow 0.2} \frac{x^2+5.8x-1.2}{x^2-4.2x+0.8} \), calculating directly at \( x = 0.2 \) might result in an indeterminate form. Here, limits provide a way to foresee how functions behave near these critical inputs. Understanding limits lays the groundwork for exploring more advanced topics, such as continuity and derivatives.
Evaluating limits is particularly significant when dealing with rational functions. It’s essential to analyze both the numerator and the denominator's behavior to predict the functional trend at the point of interest. This understanding can help students not only deal with academic problems but also apply these solutions to real-world numerical challenges.
When dealing with expressions like \( \lim_{x \rightarrow 0.2} \frac{x^2+5.8x-1.2}{x^2-4.2x+0.8} \), calculating directly at \( x = 0.2 \) might result in an indeterminate form. Here, limits provide a way to foresee how functions behave near these critical inputs. Understanding limits lays the groundwork for exploring more advanced topics, such as continuity and derivatives.
Evaluating limits is particularly significant when dealing with rational functions. It’s essential to analyze both the numerator and the denominator's behavior to predict the functional trend at the point of interest. This understanding can help students not only deal with academic problems but also apply these solutions to real-world numerical challenges.
Function Evaluation
Function evaluation is the process of determining the output of a function for particular input values. When calculating limits, function evaluation becomes important as it aids in predicting behavior without direct substitution into potentially problematic points.
In our exercise, we evaluate the function \( f(x) = \frac{x^2 + 5.8x - 1.2}{x^2 - 4.2x + 0.8} \) at values close to 0.2, such as 0.19 and 0.21. This provides an approximation of the limit as these input values do not result in indeterminate forms.
By substituting values into the function from both left and right sides of the point 0.2, we are able to monitor the output changes. Not only does this indicate the function's direction, but it also facilitates a clearer understanding of how the function behaves just before and after the critical point. These evaluations offer a practical experience of numerical analysis and solidify the concepts of limit approximation.
In our exercise, we evaluate the function \( f(x) = \frac{x^2 + 5.8x - 1.2}{x^2 - 4.2x + 0.8} \) at values close to 0.2, such as 0.19 and 0.21. This provides an approximation of the limit as these input values do not result in indeterminate forms.
By substituting values into the function from both left and right sides of the point 0.2, we are able to monitor the output changes. Not only does this indicate the function's direction, but it also facilitates a clearer understanding of how the function behaves just before and after the critical point. These evaluations offer a practical experience of numerical analysis and solidify the concepts of limit approximation.
Approximate Limits
Approximate limits are used when an exact solution to a limit is difficult or unnecessary, and instead, an estimated value is sufficient. In calculus, when evaluating limits, such approximation provides a practical insight into the function's behavior.
We use nearby values to understand the function's trend near a potential indeterminate point. For example, considering values like 0.199 and 0.201, which are close to 0.2, helps pinpoint the function's tendency.
This method is beneficial in scenarios where precise calculations aren't possible, or where estimating a value expedites the understanding of the functional trend. Approximate limits are similar to zooming in on a graph to see where the plot heads towards as it approaches a particular point.
Verifying the consistency of these approximate values from both sides of the target point is key. It helps ensure that the limit is stable and close to a particular numerical value, which is essential for deducing the correct approximate limit.
We use nearby values to understand the function's trend near a potential indeterminate point. For example, considering values like 0.199 and 0.201, which are close to 0.2, helps pinpoint the function's tendency.
This method is beneficial in scenarios where precise calculations aren't possible, or where estimating a value expedites the understanding of the functional trend. Approximate limits are similar to zooming in on a graph to see where the plot heads towards as it approaches a particular point.
Verifying the consistency of these approximate values from both sides of the target point is key. It helps ensure that the limit is stable and close to a particular numerical value, which is essential for deducing the correct approximate limit.
