Verified Answer :

  • Perform polynomial long division on the function \(f(x) = \frac{9x^2 + 36x + 41}{3x + 5}\) to find the quotient.
  • Divide \(9x^2 + 36x + 41\) by \(3x + 5\) to get the quotient \(3x + 7\) with a remainder of 6.
  • Write the function in the form \(f(x) = 3x + 7 + \frac{6}{3x + 5}\).
  • Observe that as \(x \to \pm \infty\), the function approaches the line \(y = 3x + 7\), yielding the final value \(\boxed{7}\).

Explanation

  1. Problem Analysis and Method Selection

To find the oblique asymptote of the function \(f(x) = \frac{9x^2 + 36x + 41}{3x + 5}\), we perform polynomial long division of the numerator by the denominator. An oblique asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator, and it is given by the quotient of this division.

  1. Performing Polynomial Division

We divide \(9x^2 + 36x + 41\) by \(3x + 5\):

  1. Divide the leading term of the numerator by the leading term of the denominator: \(\frac{9x^2}{3x} = 3x\).
  2. Multiply \(3x\) by the divisor: \(3x(3x + 5) = 9x^2 + 15x\).
  3. Subtract this from the original numerator: \((9x^2 + 36x + 41) - (9x^2 + 15x) = 21x + 41\).
  4. Divide the new leading term by the leading term of the denominator: \(\frac{21x}{3x} = 7\).
  5. Multiply \(7\) by the divisor: \(7(3x + 5) = 21x + 35\).
  6. Subtract this from the current remainder: \((21x + 41) - (21x + 35) = 6\).

Thus, the function can be rewritten as:
\(f(x) = 3x + 7 + \frac{6}{3x + 5}\).

  1. Identifying the Asymptote Constant

As \(x \to \pm \infty\), the term \(\frac{6}{3x + 5}\) approaches 0. Therefore, the function \(f(x)\) approaches the linear expression \(y = 3x + 7\). Comparing this to the given form \(y = 3x + k\), we can identify that \(k = 7\).

  1. Final Answer

The value of \(k\) in the oblique asymptote \(y = 3x + k\) is \(\boxed{7}\).

Examples

Oblique asymptotes are vital in engineering and economics, particularly when analyzing growth trends. For instance, in supply chain management, if the total cost of production follows a rational function like this one, the oblique asymptote represents the 'variable cost per unit' trend as production scales up, allowing managers to predict long-term operational costs effectively. Understanding these limits helps in making informed decisions about scaling and pricing in various industries.

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