Problem 75 - Calculus Early Transcendentals

Question

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\) where \(n\) is a positive integer and a is a real number. $$\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1}\left(\text { Hint: } x-1=(\sqrt[3]{x})^{3}-(1)^{3}\text { ). }\right.$$

Step-by-Step Solution

Answer

In evaluating the limit: $$\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1}$$ we used the given factorization formula: \(x^n-a^n=(x-a)\left(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+xa^{n-2}+a^{n-1}\right)\) Then we applied the formula to rewrite the limit in a simpler form and evaluated the result, which is: $$\lim_{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1} = \frac{1}{3}$$

Step 1: Use the factorization formula

From the hint, we know that \(x-1 = (\sqrt[3]{x})^3 - 1^3\) which can be treated as \(x^n - a^n\) where \(n=3\) and \(a=1\). Using the factorization formula, we can rewrite this as: \(x-1=(\sqrt[3]{x}-1)\left(\sqrt[3]{x^2}+\sqrt[3]{x}+1\right)\). Now we will substitute this expression in the limit.

Step 2: Substitute the factored expression

Place the factored expression into the given limit: $$\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{(\sqrt[3]{x}-1)\left(\sqrt[3]{x^2}+\sqrt[3]{x}+1\right)}$$

Step 3: Simplify the expression

Notice that we can cancel out the common term \((\sqrt[3]{x}-1)\): $$\lim _{x \rightarrow 1} \frac{1}{\sqrt[3]{x^2}+\sqrt[3]{x}+1}$$

Step 4: Evaluate the limit

Now, we can find the limit by substituting \(x=1\): $$\lim _{x \rightarrow 1} \frac{1}{\sqrt[3]{x^2}+\sqrt[3]{x}+1} = \frac{1}{\sqrt[3]{1^2}+\sqrt[3]{1}+1} = \frac{1}{1+1+1}$$ Finally, we can simplify the result: $$\frac{1}{3}$$ So, the limit is: $$\lim_{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1} = \frac{1}{3}$$

Key Concepts

Factorization Formula

The factorization formula is a mathematical tool that helps simplify expressions into more manageable parts. It is particularly useful when dealing with polynomials and their roots. In the context of calculus, it aids in limit evaluation when direct substitution is not feasible. The formula given here is:\[x^{n}-a^{n} = (x-a)(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots +xa^{n-2}+a^{n-1})\]This expression breaks down a complex polynomial difference into a simpler product of terms. The process involves recognizing patterns or similarities with the binomial theorem and applying the formula to achieve a simplified form. By doing so, terms can often be canceled or reduced, which is highly beneficial in evaluating limits or finding derivatives.

• Pattern Recognition: Notice terms like powers and constants that can fit into the formula structure.
• Simplification: Helps reduce complex polynomial differences into simple factors.

Limit Calculation

Limit calculation in calculus is about finding the value that a function approaches as the input approaches some value. In exercises involving indeterminate forms like \( \frac{0}{0} \), the factorization formula can be employed to simplify expressions.For example, to evaluate \[\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1}\]we can factor the denominator using the hint that \[x-1 = (\sqrt[3]{x})^3 - 1^3\]Substituting this into the expression simplifies it significantly, allowing us to eliminate terms that cause the indeterminate form. Once simplified, substituting the limit point directly often yields a finite result.

• Simplification: Identifying and canceling common factors avoids undefined expressions.
• Substitution: Plug the limiting value directly after simplification for easy evaluation.

Cubic Roots

Cubic roots, also known as cube roots, are numbers that multiply by themselves three times to yield an original number. Often expressed as \(\sqrt[3]{x}\), understanding their properties is crucial when working on problems involving polynomial roots.In calculus, cubic roots can introduce expressions that need careful simplification, especially in limits and derivatives. A crucial idea is that manipulating these roots with algebraic identities or factorization helps in tackling complex limit calculations.For instance, given\[x-1 = (\sqrt[3]{x})^3 - 1^3\]we can utilize it as a factoring tool to simplify the cubic root expressions involved.

• Algebraic Manipulation: Reduces complex root expressions into simpler forms.
• Application in Limits: Helps in finding limits where direct substitution is challenging.