Evaluating The Limit Of Radical Functions A Step By Step Guide

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In this article, we will delve into the process of evaluating a specific limit problem involving radical functions. The problem at hand is to determine the limit of the expression

lim (x→0) [√(x² - x + 1) - 1] / [√(1 + x) - √(1 - x)]

This is a quintessential calculus problem that requires a strong understanding of limit concepts, algebraic manipulation, and techniques for handling indeterminate forms. The expression initially presents an indeterminate form of 0/0 as x approaches 0, necessitating the use of methods such as rationalization to simplify the expression and find the limit. This article will walk you through a step-by-step solution, offering insights and explanations along the way to ensure a clear understanding of the underlying principles.

We are tasked with finding the limit of the following function as x approaches 0:

lim (x→0) [√(x² - x + 1) - 1] / [√(1 + x) - √(1 - x)]

The challenge lies in the fact that a direct substitution of x = 0 results in the indeterminate form 0/0. This signals the need for algebraic manipulation to simplify the expression before we can evaluate the limit. To tackle this, we'll employ a technique called rationalization, which involves multiplying the numerator and denominator by their respective conjugate expressions. This method helps eliminate the square roots, making the expression more amenable to limit evaluation.

Step 1: Rationalize the Numerator

To begin, we rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(x² - x + 1) + 1. This gives us:

lim (x→0) {[√(x² - x + 1) - 1] / [√(1 + x) - √(1 - x)]} * {[√(x² - x + 1) + 1] / [√(x² - x + 1) + 1]}

Multiplying out the numerators, we get:

lim (x→0) [(x² - x + 1) - 1] / {[√(1 + x) - √(1 - x)] * [√(x² - x + 1) + 1]}

This simplifies to:

lim (x→0) (x² - x) / {[√(1 + x) - √(1 - x)] * [√(x² - x + 1) + 1]}

Step 2: Rationalize the Denominator

Next, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is √(1 + x) + √(1 - x). This yields:

lim (x→0) {(x² - x) / [√(1 + x) - √(1 - x)] * [√(x² - x + 1) + 1]} * {[√(1 + x) + √(1 - x)] / [√(1 + x) + √(1 - x)]}

Multiplying out the denominators, we get:

lim (x→0) {(x² - x) * [√(1 + x) + √(1 - x)]} / {[(1 + x) - (1 - x)] * [√(x² - x + 1) + 1]}

This simplifies to:

lim (x→0) {(x² - x) * [√(1 + x) + √(1 - x)]} / {2x * [√(x² - x + 1) + 1]}

Step 3: Simplify the Expression

We can now factor out an x from the numerator:

lim (x→0) {x(x - 1) * [√(1 + x) + √(1 - x)]} / {2x * [√(x² - x + 1) + 1]}

Cancel out the common factor of x:

lim (x→0) {(x - 1) * [√(1 + x) + √(1 - x)]} / {2 * [√(x² - x + 1) + 1]}

Step 4: Evaluate the Limit

Now that we have simplified the expression, we can directly substitute x = 0 to evaluate the limit:

lim (x→0) {(x - 1) * [√(1 + x) + √(1 - x)]} / {2 * [√(x² - x + 1) + 1]}

Substituting x = 0, we get:

{(0 - 1) * [√(1 + 0) + √(1 - 0)]} / {2 * [√(0² - 0 + 1) + 1]}

This simplifies to:

{(-1) * [1 + 1]} / {2 * [1 + 1]}

Which further simplifies to:

(-1 * 2) / (2 * 2)

-2 / 4

-1/2

Therefore, the limit is -1/2.

In conclusion, we have successfully evaluated the limit

lim (x→0) [√(x² - x + 1) - 1] / [√(1 + x) - √(1 - x)]

by employing the technique of rationalization. This method involved multiplying both the numerator and the denominator by their respective conjugates to eliminate the square roots and simplify the expression. After algebraic manipulation and simplification, we were able to directly substitute x = 0 and find the limit to be -1/2. This problem highlights the importance of algebraic techniques in evaluating limits, particularly when dealing with indeterminate forms.

Key Takeaways

  1. Rationalization: Rationalizing the numerator and denominator is a crucial technique for dealing with limits involving square roots.
  2. Indeterminate Forms: Recognizing indeterminate forms like 0/0 is essential for determining the appropriate method for limit evaluation.
  3. Algebraic Manipulation: Skillful algebraic manipulation is key to simplifying complex expressions and making them easier to evaluate.
  4. Direct Substitution: After simplifying, direct substitution can often be used to find the limit.
  5. Step-by-Step Approach: Breaking down the problem into smaller steps can make it more manageable and reduce the likelihood of errors.

This detailed solution not only provides the answer but also elucidates the process, making it a valuable resource for students and enthusiasts of calculus. Understanding these techniques is crucial for mastering limit problems and other related concepts in calculus.