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. $$\lim _{x \rightarrow 2} \frac{x^{5}-32}{x-2}$$

Short answer

Question: Find the limit of the function \(\frac{x^5 - 32}{x - 2}\) as \(x\) approaches 2. Answer: 80

Step-by-step solution

Identify the values of n and a

We can see that in the given expression, the exponent is 5, which means \(n = 5\). Also, we are taking the limit as \(x\) approaches 2, so \(a = 2\).

Apply the factorization formula

Using the formula \(x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + ... + xa^{n-2} + a^{n-1})\), let's factor the numerator \(x^5 - 32\): \((x^5 - 2^5) = (x - 2)(x^4 + x^3(2) + x^2(2^2) + x(2^3) + 2^4)\) So our expression becomes: \(\frac{(x - 2)(x^4 + 2x^3 + 4x^2 + 8x + 16)}{(x - 2)}\)

Simplify the expression

Now, we can cancel out the \((x - 2)\) terms in the numerator and the denominator: \(\frac{x^4 + 2x^3 + 4x^2 + 8x + 16}{1}\) This leaves us with the polynomial \(x^4 + 2x^3 + 4x^2 + 8x + 16\).

Evaluate the limit

We now need to find the limit of the simplified expression as \(x\) approaches 2: \(\lim_{x \rightarrow 2} (x^4 + 2x^3 + 4x^2 + 8x + 16)\) Substituting \(x = 2\) into the expression: \(2^4 + 2(2)^3 + 4(2)^2 + 8(2) + 16 = 16 + 16 + 16 + 16 + 16\) Therefore, the limit is: \(\lim_{x \rightarrow 2} \frac{x^5 - 32}{x - 2} = \boxed{80}\)

Key concepts

Factorization Formula

The factorization formula is a powerful algebraic tool used to break down complex expressions into simpler components. The formula specifically for the difference of two similar powers, whether they are squares, cubes, or higher powers, simplifies expressions and facilitates easier problem solving. For example, the formula \( x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \cdots + xa^{n-2} + a^{n-1}) \) plays a critical role in handling limits that at first glance seem to involve complex division by zero oddities.

Using this formula allows us to factor a polynomial in the numerator of a rational function, such as \( x^5 - 32 \) into simpler polynomial expressions when \( x \) approaches a certain value, like 2.

• This transformation is essential in eliminating discontinuity in the function.
• Polynomials factored this way systematically allow for terms to cancel on top and bottom, simplifying the expression significantly.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial expression to its simplest form. This usually consists of combining like terms or factoring out common components. In the context of limits, we often encounter polynomials in the numerator that seem difficult to handle.

In our problem, once we applied the factorization formula and cancelled \( (x - 2) \) from both the numerator and the denominator, we arrived at a much simpler polynomial \(x^4 + 2x^3 + 4x^2 + 8x + 16\).

• Removing the \( (x - 2) \) from both the numerator and the denominator resolves any discontinuity when evaluating the limit at \( x = 2 \).
• This step is crucial for calculating limits since it provides a continuous function that can be directly evaluated.

Further simplification allows us to evaluate the polynomial directly by substituting the approaching value, ensuring the continuity and solvability of the original expression.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to solve equations or evaluate limits. Well-practiced manipulation is fundamental in dealing with complex polynomial expressions and ensuring they become more manageable.

Applying algebraic manipulation physically to our initial limit problem, we factor, cancel, and substitute effectively using:

• Carefully applying known formulas like the factorization formula for differences of powers.
• Step-by-step polynomial simplification to remove problematic terms causing a division by zero.
• Substituting the limit value only after simplification to avoid any undefined or infinite expressions.

Thus, algebraic manipulation is more than just solving equations; it involves detailed attention to every operation on an expression to make it solvable and coherent, especially when approaching specific values through limits.