Question

For some real number \(\mathrm{k}\), the value of \(\operatorname{Lt}_{x \rightarrow-k} \frac{x^{5}+k^{5}}{x+k}\) can be (1) 50 (2) 60 (3) 70 (4) 80

Short answer

Answer: 50.

Step-by-step solution

Define the function and limit

Given the function \(\frac{x^5+k^5}{x+k}\), we are asked to find the limit as \(x\) approaches \(-k\): \(\operatorname{Lt}_{x \rightarrow -k} \frac{x^5+k^5}{x+k}\).

Remove the common factors

Using the difference of powers factorization formula, we can rewrite the expression as \(\frac{(x-k)(x^4+x^3k+x^2k^2+xk^3+k^4)}{x+k}\). Now, replace \(x\) with \(-k + h\), where \(h\) is a small value, and the expression becomes \(\frac{((-k+h)-k)((-k+h)^4+(-k+h)^3k+(-k+h)^2k^2+(-k+h)k^3+k^4)}{-k+h+k}\). This simplifies to \(\frac{h(h^4-4hk^4+k^5)}{h}\).

Simplify the expression

Divide by \(h\) to get the simplified expression: \(h^4-4hk^4+k^5\).

Take the limit as h approaches 0

Let's find the limit as \(h\) approaches 0: \(\operatorname{Lt}_{h\rightarrow 0} (h^4 - 4hk^4 + k^5)\). Since \(k^5\) is constant, it doesn't change as \(h\) approaches 0. Therefore, the limit becomes \(0 - 4\cdot 0 \cdot k^4 + k^5 = k^5\).

Compare the result to the given options

Now, we have to find the value of \(k^5\) that matches one of the given options. (1) If \(k^5 = 50\), then \(k = \sqrt[5]{50}\). (2) If \(k^5 = 60\), then \(k = \sqrt[5]{60}\). (3) If \(k^5 = 70\), then \(k = \sqrt[5]{70}\). (4) If \(k^5 = 80\), then \(k = \sqrt[5]{80}\). Since the problem states that the value of the limit can be one of the given options, we conclude that the correct answer is (1) 50 because k^5 = 50 represents the function's limit as \(x\) approaches \(-k\).

Key concepts

Difference of Powers

The concept of the difference of powers plays a notable role when dealing with limits involving polynomial expressions. Imagine you have two terms, each raised to a power, and you need to simplify this expression. That's where the difference of powers comes into play!

For instance, when simplifying an expression like \(x^5 + k^5\), these terms are treated similarly to \(a^n + b^n\). Using the rules of exponents, you can factor such expressions, especially when aiming to simplify limits in calculus.

• Identify the common powers
• Use algebraic identities for factorization
• Simplify the expression effectively

This method transforms a complex term into a more manageable form, helping students evaluate limits with ease.

Factorization

Factorization is an essential skill when working with polynomials and limits in calculus. It involves breaking down an expression into simpler terms that are multiplied together. This approach can significantly simplify the computation of a limit.

In our exercise, factorization is used to rewrite \(x^5 + k^5\) in a form that cancels out potential problematic terms that cause indeterminacy. Using the idea of polynomial division, the polynomial is split into factors that can be more easily analyzed.

• Recognize patterns in polynomial expressions
• Apply algebraic identities for simplification
• Reduce expressions to facilitate limit evaluation

With factorization, handling complicated polynomials becomes a breeze, turning intricate calculations into straightforward problem-solving steps.

Real Numbers

Real numbers are the backbone of calculus and limit theory. They include both rational and irrational numbers, forming a continuous line of values. When dealing with limits, real numbers ensure that all possible values a variable can approach are considered.

In our problem, the concept of a limit as \(x\) approaches \(-k\) implies examining values very close to \(-k\), but not exactly \(-k\). Here, the real number \(k\) represents a real value that the expression moves closely towards as \(x\) nears \(-k\). Real numbers ensure that calculus covers every point on a number line without gaps.

• Understand their role in continuity and limits
• Recognize \(k\) as a real number in polynomial expressions
• Apply them in various limit scenarios

Real numbers establish the framework within which limits operate, connecting abstract mathematical concepts to real-world applications.

Polynomial Division

Polynomial division is a technique similar to long division but for polynomials. It’s a useful tool for simplifying polynomial expressions and analyzing limits in calculus.

In this exercise, polynomial division helps break down \(x^5 + k^5\) by dividing it by \(x + k\). This method allows us to simplify the expression by canceling terms effectively, especially when approaching a specific point where direct substitution might cause a divide-by-zero error.

• Simplify complex polynomial expressions
• Identify and cancel out indeterminate forms
• Make limit evaluation more straightforward

Using polynomial division enhances your ability to manipulate algebraic expressions, turning complex problems into simpler tasks, paving the way for accurate limit calculation.