Problem 22 - Thomas Calculus

Question

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$h(x)=\frac{5 x^{8}-2 x^{3}+9}{3+x-4 x^{5}}$$

Step-by-Step Solution

Answer

As \(x \rightarrow \infty\), the limit is \(+\infty\); as \(x \rightarrow -\infty\), the limit is \(-\infty\).

Step 1: Identify Dominant Terms

The function is \(h(x) = \frac{5x^8 - 2x^3 + 9}{3 + x - 4x^5}\). The degree of the numerator is 8 (from \(5x^8\)) and the degree of the denominator is 5 (from \(-4x^5\)). The dominant term in the numerator is \(5x^8\) and in the denominator is \(-4x^5\).

Step 2: Simplify the Expression for \(x \rightarrow \infty\) and \(x \rightarrow -\infty\)

For \(x \rightarrow \pm\infty\), the behavior of \(h(x)\) is dictated by the highest-degree terms. Simplify the rational function to \(\frac{5x^8}{-4x^5} = \frac{5}{-4}x^{8-5} = -\frac{5}{4}x^3\).

Step 3: Evaluate the Limit as \(x \rightarrow \infty\)

As \(x \rightarrow \infty\), \(-\frac{5}{4}x^3\) tends to \(+\infty\) because the exponent is positive and the coefficient is negative.

Step 4: Evaluate the Limit as \(x \rightarrow -\infty\)

As \(x \rightarrow -\infty\), \(-\frac{5}{4}x^3\) tends to \(-\infty\) because \((-x)^3 = -x^3\) for negative \(x\), making the product of negative terms positive but the leading coefficient negative.

Key Concepts

Dominant Terms

When dealing with limits of rational functions, identifying the dominant terms is crucial. Rational functions are fractions where both the numerator and the denominator are polynomials. In the function we are examining, \(h(x) = \frac{5x^8 - 2x^3 + 9}{3 + x - 4x^5}\), we first observe the terms in both the numerator and the denominator.

Typically, as \(x\) gets very large (whether positive or negative), the terms with the highest degree in both the numerator and denominator will dominate the overall behavior of the function. These are our dominant terms.

• In the numerator \(5x^8\) is the highest degree term because it has the largest exponent.
• In the denominator, the term \(-4x^5\) is the dominant one for the same reason.

By focusing on these terms, we simplify the analysis of the function's behavior as \( x \rightarrow \pm \infty \). This simplifies the complexity and makes evaluating limits much more manageable.

Behavior at Infinity

The behavior of a rational function as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \) largely depends on the dominant terms. With the identified dominant terms, we can simplify the original complex rational function to its simplest form: \[h(x) = \frac{5x^8}{-4x^5} = -\frac{5}{4}x^3\]This simplification tells us how the function behaves at infinity.

When considering limits, we want to understand how this function behaves as \(x\) grows larger and larger without bound.

• If the simplified expression results in a polynomial of positive degree, like \(-\frac{5}{4}x^3\), the function will tend towards positive or negative infinity.
  
• If the degree of the numerator was less than that of the denominator, the limit would have been zero.
  
• Conversely, if they were equal, the limit would resolve to a constant ratio of the leading coefficients.
  

Understanding this behavior at infinity is crucial for evaluating limits accurately.

Evaluating Limits

Now, we apply our understanding of dominant terms and behavior at infinity to evaluate the limits as \( x \rightarrow \pm \infty \). For the expression \(-\frac{5}{4}x^3\):

• As \( x \rightarrow \infty \): The term \(-\frac{5}{4}x^3\) behaves accordingly. Since the exponent 3 is positive and the coefficient \(-\frac{5}{4}\) is negative, the function will move towards \(+\infty\). Positive infinity arises because multiplying a large positive number cubed by a negative constant results in an increasingly large negative value.
  
• As \( x \rightarrow -\infty \): Here, \((-x)^3 = -x^3\). The same reasoning applies but in reverse. The negativeness of \((-x)^3\) turns positive with negative multiplication, leading the limit to \(-\infty\).
  

Understanding these principles gives us a clear picture that as we move towards \( \pm \infty \), the dominant terms lead the behavior of \(h(x)\), allowing us to predict and determine the limits accurately.