How to Find the Limit: A Step-by-Step Solution

Introduction

In mathematics, understanding how to compute limits is crucial for many advanced topics, including calculus and differential equations. In this article, we will walk through a specific example problem: finding the limit of the expression L limx→0 ((1/(x3) - 1/3) / x). We will explore each step in detail and discuss different methods to find the limit, including algebraic simplification and L'H?pital's Rule.

Simplifying the Expression

Our goal is to simplify the expression (frac{frac{1}{x^3} - frac{1}{3}}{x}) step by step.

Simplify the numerator: The numerator is (frac{1}{x^3} - frac{1}{3}). To combine these fractions, we need a common denominator. The common denominator here is (3x^3). (frac{1}{x^3} - frac{1}{3} frac{3 - x^3}{3x^3}) (frac{3 - x^3}{3x^3} frac{3 - x^3}{3x^3} frac{-x^3}{3x^3} frac{-x^3}{3x^3}) Substitute back into the limit: Now substitute this back into the limit expression:

Step 3: Simplify the expression

(L lim_{x to 0} frac{frac{-x^3}{3x^3}}{x} lim_{x to 0} frac{-x^3}{3x^3 cdot x} lim_{x to 0} frac{-1}{3x^3}) Evaluate the limit: Finally, we directly evaluate the limit as (x to 0):

Step 4: Evaluate the limit

(L frac{-1}{3 cdot 0^3} frac{-1}{9}) Conclusion: Thus, the limit is (boxed{-frac{1}{9}})

Alternative Methods

1. L'H?pital's Rule

Another method to solve this limit is to use L'H?pital's Rule, which is particularly useful when the limit evaluates to the indeterminate form (frac{0}{0}).

L’Hopital’s Rule: [lim_{x to 0} frac{frac{1}{x^3} - frac{1}{3}}{x} lim_{x to 0} -frac{1}{x^3 cdot 3} -frac{1}{9}]

2. Concept of Limit

Sometimes the concept of a limit can be clearer if we think of it in terms of a sequence. Consider the sequence of values that the function (f(x) frac{frac{1}{x^3} - frac{1}{3}}{x}) takes as (x) approaches 0 from both sides. Since (x) never actually equals 0, the function remains computable:

(f(x) frac{-x^3}{3x^3} frac{-1}{3x^3})

(lim_{x to 0} frac{-1}{3x^3} -frac{1}{9})

3. Algebraic Manipulation

We can factor out the term (x) from the expression to simplify the calculation:

(frac{1}{x^3} - frac{1}{3} frac{3 - x^3}{3x^3})

(frac{3 - x^3}{3x^3} frac{-x^3}{3x^3} frac{-1}{3x^3})

Now, substituting this into the limit expression, we get:

(L lim_{x to 0} frac{-1}{3x^3})

(L -frac{1}{9})

This confirms the result using algebraic manipulation.

Conclusion

In conclusion, the limit of the expression L limx→0 ((1/(x3) - 1/3) / x) is (-frac{1}{9}). This problem demonstrates the importance of algebraic manipulation and different methods for approaching and solving limits in calculus.