Rational Functions & Polynomial Limits: Examples & Solutions

Hey guys! Let's dive into the fascinating world of rational functions and polynomial limits. Today, we're going to tackle some intriguing problems involving finding a rational function with a specific limit at infinity and determining a polynomial that satisfies a given limit condition. These types of problems can seem daunting at first, but with a clear understanding of the underlying concepts, we can break them down and solve them step by step. So, grab your thinking caps, and let's get started!

Finding a Rational Function with a Limit of -√3 at +[infinity]

Okay, so the first part of our challenge is to find a rational function whose limit as x approaches positive infinity is -√3. What does this actually mean? Well, a rational function is simply a function that can be expressed as the ratio of two polynomials, like this: f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials.

The limit at infinity tells us what happens to the function's value as x gets incredibly large. In our case, we want the function to approach -√3 as x goes to +[infinity]. The key to solving this lies in understanding how the degrees of the polynomials in the numerator and denominator affect the limit at infinity.

To achieve a finite non-zero limit at infinity, the degrees of the polynomials P(x) and Q(x) must be the same. Why is this? Think about it this way: if the degree of the numerator is higher, the function will tend towards infinity (or negative infinity). If the degree of the denominator is higher, the function will tend towards zero. So, to get a finite limit, the highest powers of x in both the numerator and the denominator need to "compete" with each other. This competition results in the ratio of their leading coefficients determining the limit.

In mathematical terms, if we have P(x) = a_n x^n + ... and Q(x) = b_n x^n + ... (where a_n and b_n are the leading coefficients and n is the degree), then the limit as x approaches infinity of P(x)/Q(x) is a_n / b_n. This is a crucial concept for solving limit problems involving rational functions.

Now, we want this limit to be -√3. This means we need to choose polynomials P(x) and Q(x) such that the ratio of their leading coefficients is -√3. There are infinitely many ways to do this! A simple example would be to let P(x) be -√3x and Q(x) be x. Both are polynomials of degree 1, and the ratio of their leading coefficients is indeed -√3. So, our rational function could be f(x) = -√3x / x = -√3. This is a valid, though perhaps a bit trivial, solution. Another valid solution can be f(x) = (-√3x + 5) / (x + 2). In this case the limit as x approaches infinity will be (-√3) / 1 = -√3.

We can get more creative, of course. We could choose P(x) = -√3x² + something and Q(x) = x² + something else, as long as the leading coefficients maintain the -√3 ratio. For example, f(x) = (-√3x² + 2x - 1) / (x² + x + 3) also satisfies the condition. The most important thing is to ensure that the degrees of the polynomials are the same and the ratio of their leading coefficients gives us the desired limit. Remember, there isn't just one correct answer here; there are countless rational functions that meet the criteria. The beauty of math lies in this flexibility and the ability to find multiple solutions to a single problem.

Finding a Polynomial W(x) to Satisfy a Limit Condition

The second part of our problem presents a slightly different challenge. We need to find a polynomial W(x) that satisfies the following limit condition: lim (x→-[infinity]) [(-x³ + 2x² - 1)(2x⁴ - 5x² + 2)] / W(x) = 4/3. This looks a bit more complicated, but don't worry, we can break it down.

First, let's focus on the numerator. We have the product of two polynomials: (-x³ + 2x² - 1) and (2x⁴ - 5x² + 2). To understand the behavior of this product as x approaches negative infinity, we need to determine its degree and leading coefficient. When multiplying polynomials, the degree of the resulting polynomial is the sum of the degrees of the individual polynomials. In this case, we have a polynomial of degree 3 multiplied by a polynomial of degree 4, so the resulting polynomial will have a degree of 3 + 4 = 7.

Now, let's find the leading coefficient. The leading coefficient of a product of polynomials is the product of the leading coefficients of the individual polynomials. The leading coefficient of (-x³ + 2x² - 1) is -1, and the leading coefficient of (2x⁴ - 5x² + 2) is 2. Therefore, the leading coefficient of their product is (-1) * 2 = -2. So, the numerator behaves like -2x⁷ as x approaches negative infinity. We can write the numerator as: (-x³ + 2x² - 1)(2x⁴ - 5x² + 2) = -2x⁷ + ... where the “...” represent terms of lower degree.

Now, let's consider the limit condition: lim (x→-[infinity]) [-2x⁷ + ...] / W(x) = 4/3. For this limit to exist and be equal to 4/3, the degree of W(x) must be the same as the degree of the numerator, which is 7. This is because, as we discussed earlier, the limit of a rational function at infinity depends on the ratio of the leading coefficients when the degrees of the numerator and denominator are equal. If the degree of W(x) was lower than 7, the limit would be infinite. If it was higher than 7, the limit would be zero.

So, we know that W(x) must be a polynomial of degree 7. Let's write it in the general form: W(x) = a_7 x⁷ + a_6 x⁶ + ... + a_0. Our goal is to find the coefficient a_7 such that the limit condition is satisfied. The limit as x approaches negative infinity will be the ratio of the leading coefficient of the numerator (-2) to the leading coefficient of the denominator (a_7). Therefore, we have the equation: -2 / a_7 = 4/3.

Solving this equation for a_7, we get: a_7 = -2 * (3/4) = -3/2. This tells us that the leading coefficient of W(x) must be -3/2. The other coefficients (a_6, a_5, and so on) can be any real numbers, as they won't affect the limit at infinity. We can choose them to be zero for simplicity, or we can make W(x) more complex if we wish. A simple example of W(x) that satisfies the condition is W(x) = (-3/2)x⁷.

However, to be more general, we can choose any polynomial of degree 7 with a leading coefficient of -3/2. For instance, W(x) = (-3/2)x⁷ + x⁶ - 5x² + 10 would also work. The key is that the x⁷ term has the coefficient -3/2. Understanding this principle allows us to construct a multitude of solutions.

Wrapping Up

So there you have it! We've successfully tackled two interesting problems involving rational functions and polynomial limits. We found an example of a rational function whose limit at +[infinity] is -√3, and we determined a polynomial W(x) that satisfies a given limit condition. The key takeaways are understanding how the degrees of polynomials affect limits at infinity and how to use the ratio of leading coefficients to achieve a desired limit.

Remember, these types of problems often have multiple solutions, so don't be afraid to explore different possibilities. The more you practice and the more you understand the fundamental concepts, the easier these problems will become. Keep up the great work, guys, and happy problem-solving!