Evaluating Limits and Rationalizing Expressions: A Comprehensive Guide

In this article, we will explore the methods to evaluate limits, specifically focusing on rationalizing the numerator. This technique is particularly useful in solving limit problems that involve cube roots or other irrational expressions. We will go through an illustrative example by evaluating the limit L limx→0 [(11/3x - 11/3(1-x)/x].

Introduction to the Problem

The initial problem given is as follows: limx→01)(a6 - b6)/(a3 - b3) after the substitutions 1/x a6 and 1 - x b6.

Step-by-Step Solution

We begin by substituting 1/x a6 and 1 - x b6. This implies that a6b6 2 and a6 - b6 2x.

Using the given substitutions, we can rewrite the limit expression:

[lim_{a, b to 1} frac{a^3 - b^3}{a^2 - b^2} lim_{a, b to 1} frac{a^2 ab b^2}{ab} lim_{a, b to 1} frac{111}{11} frac{3}{2}]

Binomial Approximation Method

Another method involves using binomial approximation. Since x is very small, we can approximate:

[1 x^n approx 1 nx]

Applying this, we get:

[1 x^{1/3} approx 1 frac{1}{3}x] [1 - x^{1/3} approx 1 - frac{1}{3}x]

Substituting these approximations into the limit expression:

[L lim_{x to 0} frac{1 frac{1}{3}x - (1 - frac{1}{3}x)}{x} frac{2}{3}]

L'Hopital's Rule Method

Another effective method involves L'Hopital's Rule. Since the form is 0/0, we can differentiate both the numerator and the denominator:

[lim_{x to 0} frac{frac{1}{3}(1 - x)^{-2/3} - frac{1}{3}(1 x)^{-2/3}}{1} frac{frac{1}{3} - frac{1}{3}}{1} frac{2}{3}]

Algebraic Method Using Difference of Cubes Formula

The algebraic method involves using the difference of cubes formula:

[a^3 - b^3 (a - b)(a^2 ab b^2)]

Applying this to the limit expression:

[L lim_{x to 0} frac{(1 - x)^{1/3} - (1 - x)^{1/3}}{x} cdot frac{(1 - x)^{2/3} (1 - x)^{1/3} 1}{(1 - x)^{2/3} (1 - x)^{1/3} 1} lim_{x to 0} frac{1 - (1 - x)}{x[(1 - x)^{2/3} (1 - x)^{1/3} 1]}]

[L lim_{x to 0} frac{2}{(1 - x)^{2/3} (1 - x)^{1/3} 1} frac{2}{3}]

Conclusion

In conclusion, evaluating limits involving cube roots or other irrational expressions can be approached through multiple methods. The algebraic method, binomial approximation, and L'Hopital's rule are all effective techniques. This article provides a comprehensive guide on how to solve such problems and highlights the importance of being familiar with various mathematical approaches.

Keywords: limit, algebraic method, rationalizing numerator