Problem 36 - Thomas Calculus

Question

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of \(x :\) Divide numerator and denominator by the highest power of \(x\) in the denominator and proceed from there. Find the limits in Exercises \(23-36\). $$\lim _{x \rightarrow-\infty} \frac{4-3 x^{3}}{\sqrt{x^{6}+9}}$$

Step-by-Step Solution

Answer

The limit is -3.

Step 1: Identify the highest power of x in the denominator

The first step is to identify the highest power of \(x\) found in the expression under the square root in the denominator. In this case, the expression is \(x^6 + 9\), which means the highest power of \(x\) in the denominator is \(x^6\).

Step 2: Rewrite the equation dividing by x^6

Divide each term in the numerator and the expression under the square root in the denominator by \(x^6\). Doing so gives \(\lim _{x \rightarrow -\infty} \frac{\frac{4}{x^6} - \frac{3x^3}{x^6}}{\sqrt{\frac{x^6}{x^6} + \frac{9}{x^6}}}\).

Step 3: Simplify the expression

Simplify the expression from step 2: \(\lim _{x \rightarrow -\infty} \frac{\frac{4}{x^6} - \frac{3}{x^3}}{\sqrt{1 + \frac{9}{x^6}}}\). As \(x\) approaches \(-\infty\), any term with \(x\) in the denominator will approach 0, leaving: \(\lim _{x \rightarrow -\infty} \frac{0 - 0}{\sqrt{1 + 0}}\).

Step 4: Evaluate the limit

Evaluate the limit: \(\lim _{x \rightarrow -\infty} \frac{-3}{0}\), which simplifies to \(-3\). Thus, the limit is \(-3\).

Key Concepts

Limits of functions

Limits in calculus represent the value that a function approaches as the input approaches a particular point. Specifically, for limits of functions, we often examine how a function behaves as the input reaches positive or negative infinity or some finite value.
Understanding limits is crucial because they help us grasp the behavior of functions—how they provide insight into function continuity, derivatives, and integrals.
When solving problems involving limits, a common technique is to simplify the expression to isolate or eliminate indeterminate parts. For instance, if a function has a form that leads to dividing by zero or infinity, we aim to clarify this behavior.
For rational functions, dividing both the numerator and the denominator by the highest power of the variable can help manage and solve such expressions. This often turns a complex form into a more manageable one, allowing us to directly observe the behavior at the limit.

Rational functions

Rational functions are fractions where both the numerator and the denominator consist of polynomials. These functions can behave quite differently based on their components, especially when taking limits.
Common characteristics of rational functions include:

• A polynomial in the numerator and another in the denominator.
• Possibility of vertical asymptotes—where the function heads towards infinity or negative infinity.
• Horizontal asymptotes that indicate the behavior of the function at extreme values (like infinity).

When working on the limits of rational functions at infinity, we analyze the degrees of the polynomials in both the numerator and the denominator.
If the degree in the denominator is higher, this typically implies the function limit approaches zero, as the denominator grows much faster. If the degrees are equal, the limit will usually be the ratio of the leading coefficients.

Negative powers of x

Negative powers of variables might seem challenging, but they're essential in comprehending rational functions and their limits.
A negative power of a variable, such as \(x^{-n}\), is equivalent to \((1/x^n)\). This setup indicates that as \(x\) becomes significantly large—either positively or negatively—the value of such expressions becomes quite small and tends towards zero.
In calculating limits, especially when dividing by negative or fractional powers, understanding this behavior allows us to simplify expressions effectively. As negative powers convert to fractions, they often help reduce complex expressions to manageable numbers, paving the way for straightforward limit evaluation.
This approach is frequently used in limit calculations, where dividing by the variable's highest power in the expression aids in understanding the limit's behavior as it approaches infinity or negative infinity.