Problem 16 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow \infty} \frac{3 x^{3}+2}{9 x^{3}-2 x^{2}+7} $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches infinity of \(\frac{3x^{3}+2}{9x^{3}-2x^{2}+7}\) is \(\frac{1}{3}\).

Step 1: Observing and Simplifying

Observe that the highest degree in both the numerator and denominator is 3. Since they hold the same degree, the limit as \(x\) approaches infinity of \(\frac{3x^{3}+2}{9x^{3}-2x^{2}+7}\) will be the ratio of the coefficients of the term \(x^{3}\) in the numerator and denominator which are 3 and 9 respectively.

Step 2: Calculating the Limit

To calculate this limit, we just divide the coefficients 3 and 9, thus getting \(\frac{3}{9} = \frac{1}{3}\)

Key Concepts

Limits at Infinity

When you see the term "limit at infinity," you are dealing with what happens to a function as the input value approaches infinity or negative infinity.
In simpler terms, imagine what the function does as the input gets really, really large (or really negative).
Here, your goal is often to find a horizontal asymptote, which is a line the graph of the function approaches but never actually touches.

• For many functions, as the input increases or decreases indefinitely, the output settles into a certain pattern or constant value.
• This constant value, if it exists, can be seen as the function's limit at infinity.
• It helps predict the long-term behavior of the function.

Understanding limits at infinity is crucial because it allows you to describe the boundary behavior of functions, letting you peek into their far-reaching tendencies. In our example, we are finding what happens to the function\[\lim _{x \rightarrow \infty} \frac{3 x^{3}+2}{9 x^{3}-2 x^{2}+7}\]as \(x\) grows very large.

Rational Functions

A rational function is simply a fraction where both the numerator and the denominator are polynomials.
These functions are quite common in calculus and can lead to interesting behavior at their limits.

• They are expressed in the form \(\frac{P(x)}{Q(x)}\) where both \(P(x)\) and \(Q(x)\) are polynomials.
• The degrees of these polynomials, meaning the highest power of \(x\), play a vital role in determining the behavior of the function.
• When exploring limits at infinity, the degrees of the polynomials tell us where to focus our attention.

In contexts like our original problem, these factors become the main criteria to decide the function's limit. When both the numerator and the denominator have the same degree, the limit is the ratio of these leading coefficients. That's how we know the limit approaches \(\frac{1}{3}\) for the problem presented.

Polynomial Division

Polynomial division is a helpful tool when working with rational functions.
While it's similar to long division in arithmetic, this process is geared towards simplifying expressions involving polynomials.

• A crucial use of polynomial division is simplifying a function to find its behavior as \(x\) becomes very large or very small.
• It helps in isolating the leading terms of the polynomials both in the numerator and denominator.
• This is because the higher degree terms dominate the behavior of the polynomial when \(x\) is large.

In our specific example, by conducting polynomial division, we learned that the powers beyond the highest degree can be neglected in terms of determining the limit.
This simplifies the expression and makes it clear that taking the leading coefficients gives us the limit of \(\frac{1}{3}\) for the rational function in the exercise.