Question

\(\operatorname{Lt}_{\mathrm{x} \rightarrow \infty} \frac{(\mathrm{x}-5)(\mathrm{x}+7)}{(\mathrm{x}+2)(5 \mathrm{x}+1)}=\) (1) \(-\frac{1}{5}\) (2) 25 (3) 5 (4) \(\frac{1}{5}\)

Short answer

Short Answer: \(\frac{1}{5}\)

Step-by-step solution

Identify the degrees of the polynomials in the numerator and the denominator

In the given function, the numerator is \((x-5)(x+7)\) and the denominator is \((x+2)(5x+1)\). By expanding the expressions, we have: Numerator: \((x-5)(x+7) = x^2 + 2x - 35\) Denominator: \((x+2)(5x+1) = 5x^2 + x + 2\) The degrees of both polynomials are: degree of the numerator = 2 and degree of the denominator = 2.

Apply the rule for limits when the degrees of the numerator and the denominator are equal

When the degrees of the numerator and the denominator are equal, the limit of the rational function as x approaches infinity is the ratio of the leading coefficients. In this case, the leading coefficients are 1 for the numerator (coefficient of \(x^2\)) and 5 for the denominator (coefficient of \(5x^2\)). Therefore, the limit is the ratio of these leading coefficients: \(\operatorname{Lt}_{\mathrm{x} \rightarrow \infty} \frac{(\mathrm{x}-5)(\mathrm{x}+7)}{(\mathrm{x}+2)(5 \mathrm{x}+1)}= \frac{1}{5}\) The correct answer is (4) \(\frac{1}{5}\).

Key concepts

Limit of a Rational Function

Understanding the limit of a rational function involves analyzing the behavior of a function as the variable approaches a certain value. In the context of our exercise, we're interested in the behavior as the variable x approaches infinity. This is represented by the notation \(\operatorname{Lt}_{\mathrm{x} \rightarrow \infty}\) before the given rational function. To determine this limit, we need to consider the degrees of the polynomials and their leading coefficients. When the degrees of the numerator and denominator are equal, as is the case here, the limit simplifies to the ratio of their leading coefficients, offering a clear view of the function's end behavior without complicated calculations. This concept is essential for predicting trends in functions and is a foundational element in calculus.

Polynomial Degree

The polynomial degree is a critical aspect in determining the behavior of a rational function, especially as the variable approaches infinity or negative infinity. In simple terms, the degree of a polynomial is the highest power of the variable x in its expression. For instance, in our example's numerator \(x^2 + 2x - 35\), the highest power of x is 2, thus it is a polynomial of degree 2. Similarly, the denominator \(5x^2 + x + 2\) is also of degree 2. The degree helps in predicting how the function behaves as x becomes very large. If the degrees of the polynomials in the numerator and denominator are equal, the rational function simplifies significantly when considering limits at infinity, by just looking at the leading coefficients and their ratio. This is vital for students to master as it applies to a wide range of problems in calculus.

Leading Coefficients

Leading coefficients play a pivotal role when evaluating the limit of rational functions with equal polynomial degrees. They are the coefficients of the term with the highest power in a polynomial. In our exercise, the leading coefficient of the numerator \(x^2 + 2x - 35\) is 1, the coefficient of \(x^2\). For the denominator \(5x^2 + x + 2\), it is 5, the coefficient of \(5x^2\). When the variable x grows without bounds, these leading coefficients dictate the ratio that will represent the limit of the rational function. This concept may seem simple, but it has a huge impact on the outcome of limits and is therefore a fundamental concept for students to understand. It is also the reason why the final answer to our problem is \(\frac{1}{5}\), simplifying a potentially complex procedure into a straightforward calculation.