Problem 30 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow-\infty} \frac{x(x-3)}{7-x^{2}} $$

Step-by-Step Solution

Answer

The limit is \(-1\).

Step 1 - Identify the Highest Power of x

To find the limit as \( x \) approaches \(-\infty\), start by identifying the highest power of \( x \) in both the numerator and the denominator of the given function. Here, the highest power of \( x \) in the numerator is \( x^2 \), and in the denominator, it is also \( x^2 \).

Step 2 - Divide by the Highest Power of x

Divide both the numerator and the denominator by \( x^2 \) to simplify the expression. This gives: \[ \frac{x(x-3)}{7-x^2} = \frac{x \cdot x - 3x}{x^2 \left( \frac{7}{x^2} - 1 \right)} \].

Step 3 - Simplify the Expression

Simplify the expression obtained in the previous step: \[ \frac{x^2 - 3x}{7 - x^2} = \frac{1 \cdot x^2/x^2 - 3x/x^2}{7/x^2 - x^2/x^2} \]. This yields: \[ \frac{1 - \frac{3}{x}}{\frac{7}{x^2} - 1} \].

Step 4 - Evaluate the Limit

As \( x \) approaches \(-\infty\), \( \frac{3}{x} \) approaches 0 and \( \frac{7}{x^2} \) also approaches 0. Therefore, the expression simplifies to: \[ \lim _{x \rightarrow -\infty} \frac{1 - 0}{0 - 1} = \lim _{x \rightarrow -\infty} \frac{1}{-1} = -1 \].

Key Concepts

limits at infinity

When dealing with limits at infinity, we aim to find the behavior of a function as the variable grows extremely large or extremely small. Approaching infinity, whether it is positive or negative, helps us understand the end behavior of functions. In this exercise, we are focusing on limits as \(x\) approaches \(-\but- infty\). Signals such as positive or negative infinity indicate whether values grow without bounds or diminish indefinitely.

highest power of x

Identifying the highest power of \( x \) is a critical step in solving limits at infinity. The highest power dominates the behavior of the function for very large or very small \( x \). In the given exercise, the highest power of \( x \) in both the numerator and the denominator is \( x^2 \). By focusing on the highest power, we can simplify the function and make it easier to evaluate the limit.

simplification of rational expressions

Simplifying rational expressions involves reducing the function to its simplest form by dividing all terms by the highest power of \( x \). This method allows us to eliminate negligible terms as \( x \) approaches infinity or negative infinity. The given problem simplifies to:

• Multiply out and divide each term by \( x^2 \).


• Simplify the fractions to identify dominant coefficients.

This results in the simplified expression \( \frac{1 - \frac{3}{x}}{\frac{7}{x^2} - 1}\).

approaching infinity

As \(x\) approaches infinity, some terms in the simplified expression tend to zero. Evaluating these limits helps find the behavior of the original expression.
For the given limit:

• \(\frac{3}{x} \) becomes 0 as \(x\) grows larger.
• \(\frac{7}{x^2} \) also tends to 0.

Thus, the expression simplifies as:
\( \frac{1 - 0}{0 - 1} = \frac{1}{-1} = -1 \).
Therefore, the limit is \(-1\).