Question

Useful factorization formula 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. $$\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}, \text { for any positive integer } n$$

Short answer

**Question:** Find the limit of the expression $\frac{x^n - a^n}{x - a}$ as x approaches a, using the given factorization formula for any positive integer n and real number a: $$x^{n}-a^{n}=(x-a)(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1})$$ **Answer:** The limit of the expression $\frac{x^n - a^n}{x - a}$ as x approaches a is: $$\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a} = n a^{n-1}$$ for any positive integer n.

Step-by-step solution

Identify the factorization formula

Here, we are given a factorization formula for any positive integer n and real number a. $$x^{n}-a^{n}=(x-a)(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1})$$

Apply the factorization formula to the given expression

Let's first apply the given factorization formula to the expression in the limit. We rewrite the given expression using the formula: $$\frac{x^{n}-a^{n}}{x-a} = \frac{(x-a)(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1})}{x-a}$$

Simplify the expression

Now, we can simplify the expression by canceling out the (x-a) terms in the numerator and denominator: $$\frac{(x-a)(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1})}{x-a}=x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1}$$

Evaluate the limit of the simplified expression

Next, we find the limit of the simplified expression as x approaches a: $$\lim _{x \rightarrow a} \left(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)$$

Substitute x with a in the expression

Since the expression is continuous and defined for x=a, we can directly substitute x with a: $$\lim _{x \rightarrow a} \left(x^{n-1}+x^{n-2}a+x^{n-3}a^{2}+\cdots+x a^{n-2}+a^{n-1}\right) = a^{n-1}+a^{n-2}a+a^{n-3}a^{2}+\cdots+a a^{n-2}+a^{n-1}$$

Simplify the result

Finally, let's simplify this expression: $$a^{n-1}+a^{n-1}+a^{n-1}+\cdots+a^{n-1}+a^{n-1} = n a^{n-1}$$ So, the result of the limit is: $$\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a} = n a^{n-1}$$ for any positive integer n.

Key concepts

Factorization

Factorization is a crucial algebraic method often used to simplify mathematical expressions. It involves breaking down a complex expression into simpler components, known as factors. In the context of polynomials, we want to express a polynomial as a product of its root factors. This process simplifies both computation and understanding of the polynomial's properties.

The original exercise utilizes a factorization formula for expressions in the format of \(x^n - a^n\). Specifically, it factors this expression into two parts: \((x-a)\) and \(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \dots + a^{n-1}\).

Let's see how it works:

• The first factor, \((x-a)\), is straightforward and represents one root.
• The second factor is a more complex polynomial comprising terms that decrease in powers of \(x\) and increase in powers of \(a\).

This factorization is handy when you want to simplify expressions involving polynomial divisions, as we've seen in the limit exercise where the \((x-a)\) terms cancel out neatly.

Limits

In calculus, limits are fundamental in assessing the behavior of functions as they approach a specific point. Understanding limits helps us grasp concepts such as continuity, rates of change, and even the foundation for derivatives and integrals.

In the exercise, the goal is to calculate \(\lim_{x \to a} \frac{x^n - a^n}{x-a}\). Without factorization, direct substitution would yield an undefined expression due to 0/0.

Here's how we handle such calculations:

• Use algebraic manipulation (factorization, in this case) to resolve indeterminate forms.
• Once simplified, substitute the limit value to evaluate the expression further.

This approach not only resolves the potential for division by zero but also illustrates how functions behave at critical points.

Polynomials

Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents of variables. They form the building blocks of more complex mathematical functions and models.

In the original exercise, the expression \(x^n - a^n\) is a polynomial. Factoring polynomials effectively deconstructs them into products that are easier to analyze, compute, and understand.

A few key characteristics of polynomials include:

• They can be classified by degree, which is the highest power of the variable in the expression.
• Roots (solutions where the polynomial equals zero) are central to understanding their behavior and graph.
• They are used extensively in calculus, physics, engineering, and various scientific fields for modeling natural phenomena.

Recognizing polynomials and understanding how to manipulate them through techniques like factorization can significantly enhance the process of solving mathematical problems efficiently.