Evaluating Limits In Calculus A Comprehensive Guide

In the fascinating realm of calculus, limits form the bedrock upon which many advanced concepts are built. Understanding limits is crucial for grasping continuity, derivatives, and integrals. This article delves into the evaluation of two specific limits, providing a step-by-step analysis and highlighting key principles. We will explore how to approach limits involving absolute values and rational functions as x approaches a specific value or infinity. Mastering these techniques is essential for any student venturing into calculus and beyond.

Problem 38: Limit as x approaches 1 from the left

Problem Statement:

Evaluate $\lim _{x \rightarrow 1^{-}} \frac{|x-1|}{x-1}$.

Understanding the Problem

This problem involves finding the limit of a function that includes an absolute value as x approaches 1 from the left (denoted as 1⁻). The absolute value function, denoted by |x|, returns the non-negative value of x. Therefore, |x| = x if x ≥ 0 and |x| = -x if x < 0. The key to solving this limit lies in understanding how the absolute value function behaves as x approaches 1 from values less than 1.

Step-by-Step Solution

1. Analyze the Absolute Value: When x approaches 1 from the left (i.e., x < 1), the expression (x - 1) is negative. Therefore, |x - 1| can be rewritten as -(x - 1) because the absolute value of a negative number is its negation. This is a critical step because it allows us to remove the absolute value, which often complicates limit evaluations. Understanding this behavior of absolute values is crucial in calculus.
2. Rewrite the Function: Replace |x - 1| with -(x - 1) in the limit expression. The function then becomes:$\frac{|x-1|}{x-1} = \frac{-(x-1)}{x-1}$
3. Simplify the Expression: The (x - 1) terms in the numerator and denominator cancel out, provided that x ≠ 1. Since we are considering the limit as x approaches 1, but not actually equal to 1, this cancellation is valid. After cancellation, the expression simplifies to:$\frac{-(x-1)}{x-1} = -1$
4. Evaluate the Limit: Now that we have simplified the function to a constant, the limit as x approaches 1 from the left is simply the constant value. Therefore:$\lim _{x \rightarrow 1^{-}} \frac{|x-1|}{x-1} = \lim _{x \rightarrow 1^{-}} -1 = -1$

Conclusion for Problem 38

The limit of the function as x approaches 1 from the left is -1. Therefore, the correct answer is D. -1. This problem emphasizes the importance of understanding the behavior of the absolute value function and how it affects the limit calculation. By correctly interpreting the absolute value, we were able to simplify the expression and find the limit.

Problem 39: Limit as x approaches infinity

Problem Statement:

Evaluate $\lim _{x \rightarrow \infty} \frac{4-x^2}{4 x^2-x-2}$.

Understanding the Problem

This problem requires us to find the limit of a rational function as x approaches infinity. A rational function is a function that can be expressed as the quotient of two polynomials. When evaluating limits at infinity, we are interested in the function's behavior as x becomes extremely large. A common technique for such problems involves dividing both the numerator and the denominator by the highest power of x that appears in the denominator. This approach simplifies the expression and allows us to determine the limit more easily. Understanding how to manipulate rational functions is crucial in various areas of calculus and analysis.

Step-by-Step Solution

1. Identify the Highest Power of x in the Denominator: In the given function, the highest power of x in the denominator is x². This term will guide our simplification process.
2. Divide Numerator and Denominator by x²: Divide each term in both the numerator and the denominator by x². This step is crucial as it helps to simplify the expression and make the limit evaluation more straightforward. This gives us:$\lim _{x \rightarrow \infty} \frac{\frac{4}{x^2}-\frac{x2}{x^2}}{\frac{4x2}{x^{2}-\frac{x}{x}2}-\frac{2}{x^2}}$
3. Simplify the Expression: After dividing, simplify the fractions. This results in:$\lim _{x \rightarrow \infty} \frac{\frac{4}{x^{2}-1}{4-\frac{1}{x}-\frac{2}{x}2}}$
4. Evaluate the Limit: As x approaches infinity, terms of the form c/xⁿ, where c is a constant and n is a positive integer, approach 0. Therefore, as x goes to infinity, 4/x² approaches 0, 1/x approaches 0, and 2/x² approaches 0. This simplifies the limit to:$\lim _{x \rightarrow \infty} \frac{0-1}{4-0-0} = \frac{-1}{4}$

Conclusion for Problem 39

The limit of the function as x approaches infinity is -1/4. However, this result is not among the provided options (A. -2, B. 1, C. 2, D. Discussion category). Therefore, there might be an error in the given options or the problem statement itself, if we are forced to select an answer from the options then the closest answer would be A. -2 since it is the only negative option available. This highlights the importance of carefully checking the provided options and the problem statement for any discrepancies.

Evaluating limits is a fundamental skill in calculus. Through the analysis of these two problems, we have explored different techniques for evaluating limits, including dealing with absolute value functions and rational functions at infinity. For absolute value functions, it is crucial to understand their piecewise nature and how they behave around the point of interest. For rational functions at infinity, dividing by the highest power of x in the denominator is a powerful technique. Mastering these techniques will greatly aid in understanding more complex calculus concepts. Remember to always carefully analyze the problem statement and options, ensuring accuracy in your solutions.