Question

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+a x^{n-2}+a^{2} x^{n-3}+\cdots+a^{n-2} x+a^{n-1}\right)\) where \(n\) is a positive integer and a is a real number. $$\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}$$

Short answer

Question: Calculate the limit of the following expression: \(\lim_{x \rightarrow a} \frac{x^5 - a^5}{x-a}\) Answer: \(5a^4\)

Step-by-step solution

Identify expression in the formula

First, we need to identify the expression in the given limit that fits the formula. In our case, that expression is \(x^5 - a^5\). The value of \(n\) for this expression is 5.

Apply the factorization formula

Next, using the factorization formula, we will factor the expression \(x^5 - a^5\). Here, we have \(x^5 - a^5 = (x-a)(x^4 + ax^3 + a^2x^2 + a^3x + a^4)\).

Substitute the factored expression into the limit

Now, we will substitute the factored expression into the given limit: $$\lim_{x \rightarrow a} \frac{x^5 - a^5}{x-a} = \lim_{x \rightarrow a} \frac{(x-a)(x^4 + ax^3 + a^2x^2 + a^3x + a^4)}{x-a}$$

Simplify the limit

Observe that we can cancel out the \((x-a)\) terms: $$\lim_{x \rightarrow a} \frac{(x-a)(x^4 + ax^3 + a^2x^2 + a^3x + a^4)}{x-a} = \lim_{x \rightarrow a} (x^4 + ax^3 + a^2x^2 + a^3x + a^4)$$

Evaluate the limit

Now, since the limit is continuous and well-defined, we can plug in \(x = a\) and obtain the limit: $$\lim_{x \rightarrow a} (x^4 + ax^3 + a^2x^2 + a^3x + a^4) = a^4 + a(a^3) + a^2(a^2) + a^3(a) + a^4$$

Simplify the result

Finally, simplify the expression to get the result: $$a^4 + a^4 + a^4 + a^4 + a^4 = 5a^4$$ The limit is \(\boxed{5a^4}\).

Key concepts

Limit calculation

Understanding limits is fundamental in calculus. A limit describes the value that a function approaches as the input (or variable) approaches a certain point. In this exercise, we look at the limit of a polynomial expression, specifically, \( \lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a} \).
When you first encounter a limit, it may seem like we can simply substitute \(x = a\) into the expression. However, this can lead to indeterminate forms like \( \frac{0}{0} \), where both the numerator and denominator become zero. Such forms require special techniques, like algebraic manipulation to resolve.
In this scenario, we're using factorization to find a valid form. Once we cancel any common terms, we can substitute \(x = a\) directly to evaluate the limit without causing indeterminacy. This technique, in general, is vital for calculating limits of expressions that initially appear problematic.

Factorization formula

The factorization formula is a crucial tool when dealing with polynomial expressions, especially during limit calculations.
For the given expression \( x^{5} - a^{5} \), the formula \( x^{n}-a^{n}=(x-a)(x^{n-1}+a x^{n-2}+a^{2} x^{n-3}+\cdots+a^{n-2}x+a^{n-1}) \) allows us to break it down.
Using this formula, we can factor \( x^{5} - a^{5} \) into the product \((x-a)(x^4 + ax^3 + a^2x^2 + a^3x + a^4)\).
The factorization makes it possible to cancel the \((x-a)\) term in the denominator, simplifying the expression and making the limit calculation straightforward.
This factorization not only assists in limit calculations but is also useful for solving polynomial equations and finding roots.

Polynomial expressions

Polynomial expressions are a common type of algebraic expression that consist of variables raised to whole number powers, each multiplied by a coefficient, and added or subtracted together.
In this exercise, the expression \( x^{5} - a^{5} \) is a specific polynomial of degree 5. It involves terms like \(x^5, a^5\), and their interaction is explored using factorization and limits.
Understanding polynomials involves recognizing terms, degrees, and how polynomials can be manipulated. Operations like addition, subtraction, and especially multiplication and division, often employ techniques such as factorization.
These concepts might initially appear abstract, but they are vital for calculus and various applications, including physics and engineering. By practicing with polynomial expressions, students develop mathematical intuition that will be invaluable in more advanced studies. Grasping these fundamentals prepares students for solving complex problems involving more intricate relationships between variables.