91Ó°ÊÓ

Rational functions play a crucial role in calculus, particularly when considering limits at infinity. A rational function is any function that can be expressed as the ratio of two polynomials. In our exercise, the function \[\lim_{t \rightarrow +\infty} \frac{6-t^{3}}{7t^{3}+3}\]is an example of a rational function. Here:

• Numerator: \(6 - t^3\)
• Denominator: \(7t^3 + 3\)

Rational functions can exhibit a variety of behaviors as the variable approaches infinity. Some become constant, others grow without bound, and some may even oscillate. Understanding these behaviors is key to mastering calculus limits. The degree of the polynomials in the numerator and the denominator heavily influences these behaviors.

When evaluating limits of rational functions at infinity, polynomial division can become a handy tool. We often look at the polynomial with the highest degree term in both the numerator and denominator.

In our example, we use the highest degree term \(t^3\) to simplify the expression. We factor \(t^3\) from both the numerator \(6 - t^3\) and the denominator \(7t^3 + 3\):
\[\frac{t^3(-1 + \frac{6}{t^3})}{t^3(7 + \frac{3}{t^3})}\]This step simplifies significantly because \(t^3\) is present in both parts, allowing us to cancel it out, leading to a simpler function to evaluate:\[\frac{-1 + \frac{6}{t^3}}{7 + \frac{3}{t^3}}\]Evaluating this simpler form as \(t\) approaches infinity helps in determining the limit.

The leading coefficient in a polynomial is the coefficient of the term with the highest degree, and it's pivotal in determining the behavior of a rational function at infinity. In our exercise:

• In the numerator \(6 - t^{3}\), the leading coefficient is \(-1\), related to the term \(-t^3\).
• In the denominator \(7t^3 + 3\), the leading coefficient is \(7\), associated with the term \(7t^3\).

When we evaluated the limit as \(t\) approaches infinity, the lower degree terms diminish in influence. Only the leading terms matter. As lower degree terms like \(\frac{6}{t^3}\) and \(\frac{3}{t^3}\) approach zero, they are negligible. Therefore, the limit simplifies to being solely dependent on the leading coefficients:
\[\frac{-1}{7}\]Understanding the impact of leading coefficients streamlines the process of finding limits for rational functions.