Problem 17 - Calculus Early Transcendentals

Question

\(15-36\) Find the limil. $$ \lim _{x \rightarrow-\infty} \frac{1-x-x^{7}}{2 x^{2}-7} $$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Identify the Dominant Terms

To find the limit as \(x\) approaches \(-\infty\), we first identify the dominant term in both the numerator \(1 - x - x^7\) and the denominator \(2x^2 - 7\). As \(x\) becomes very large and negative, the terms \(-x^7\) in the numerator and \(2x^2\) in the denominator dominate.

Step 2: Simplify the Expression

Since \(-x^7\) and \(2x^2\) dominate, divide both the numerator and the denominator by \(x^7\), the highest degree term: \[ \frac{1-x-x^7}{x^7} \quad \text{and} \quad \frac{2x^2 - 7}{x^7} \]. This simplifies to \[ \frac{x^7(1 - \frac{x}{x^7} - \frac{1}{x^7})}{x^7} \] in the numerator and \[ \frac{x^7(\frac{2x^2}{x^7} - \frac{7}{x^7})}{x^7} \] in the denominator.

Step 3: Evaluate the Simplified Expression

Evaluate each term as \(x\) approaches \(-\infty\):- Numerator: \( -1 \) because \(1/x^7\) and \(x/x^7\) tend to 0.- Denominator: \(0\) since \(2/x^5\) and \(-7/x^7\) tend to 0.The expression becomes \(-\frac{1}{0}\), which indicates the behavior as matching the sign of \(-1\), leading to \(-\infty\).

Step 4: Conclude the Solution

Thus, \( \lim _{x \rightarrow-\infty} \frac{1-x-x^{7}}{2 x^{2}-7} = -\infty \). This indicates that as \(x\) approaches \(-\infty\), the function diverges towards \(-\infty\).

Key Concepts

Dominant Term

When we talk about the dominant term in an algebraic expression, we're focusing on the term with the highest power of the variable. As we explore limits involving infinity, identifying this term becomes crucial. Why? Because, as the variable approaches infinity or negative infinity, the term with the highest degree largely determines the behavior of the entire expression.

Consider the given function, \( rac{1-x-x^7}{2x^2-7} \).

• In the numerator, as \(x\) becomes very large and negative, the term \(-x^7\) dominates because it has the highest degree, which is 7.
• In the denominator, the term \(2x^2\) dominates with a degree of 2.

Understanding these dominant terms helps us to simplify complex expressions, focusing on the terms that significantly influence the value of the function as \(x\) approaches negative infinity.

Limit to Infinity

The concept of 'Limit to Infinity' deals with finding the value that a function approaches as the input values become either very large positively or very large negatively. In this exercise, we're exploring what happens as \(x\) approaches \(-\infty\).

To find this limit, note that the dominant terms \(-x^7\) and \(2x^2\) from the numerator and denominator respectively drive the function's behavior. As \(x\) approaches \(-\infty\), the expression \(\frac{1-x-x^7}{2x^2-7}\) simplifies significantly.

• The term \(-x^7\) in the numerator becomes very large negatively.
• Since \(x^7\) grows faster than \(x^2\), the denominator's terms approach zero.

This results in the entire fraction nearing \(-\infty\), as there is no balancing term to counteract the effect of the negative numerator and near-zero denominator. Hence, the limit of the function as \(x\) tends to \(-\infty\) is \(-\infty\).

Simplification of Algebraic Expressions

Simplification is a key step when finding limits, especially at infinity. By breaking down and reducing expressions, we make them easier to analyze. In our problem, the expression \(\frac{1-x-x^7}{2x^2-7}\) was simplified by factoring out the dominant terms.

The approach involves dividing the entire expression by the highest power of \(x\) identified among the terms, which is \(x^7\) in this case.

• For the numerator, \(\frac{1-x-x^7}{x^7}\) simplifies to \(-1+\frac{x}{x^7}+\frac{1}{x^7}\).
• For the denominator, \(\frac{2x^2-7}{x^7}\) becomes \(\frac{2}{x^5}-\frac{7}{x^7}\).

As \(x\) approaches \(-\infty\), terms like \(\frac{1}{x^7}\) or \(\frac{x}{x^7}\) vanish because they tend to zero, leaving a more straightforward expression to evaluate. Effective simplification allows clearer insight into the limit behavior of the function, particularly at infinity.