Problem 24 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to -\infty}\frac{\sqrt{1 + 4x^6}}{2 - x^3} \)

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Identify the dominating terms

When evaluating limits as \( x \to -\infty \), it is useful to identify the terms with the highest power of \( x \) in the numerator and the denominator. Here, in the numerator \( \sqrt{1 + 4x^6} \), the term \( 4x^6 \) dominates as \( x \to -\infty \). In the denominator \( 2 - x^3 \), the term \( -x^3 \) dominates.

Step 2: Simplify the expression

Factor out the dominating terms from both the numerator and the denominator.\[ \frac{\sqrt{1 + 4x^6}}{2 - x^3} = \frac{\sqrt{x^6(1 + \frac{4}{x^6})}}{-x^3(1 - \frac{2}{x^3})}\]Thus, the expression simplifies to:\[ \frac{x^3\sqrt{1 + \frac{4}{x^6}}}{-x^3(1 - \frac{2}{x^3})} \]

Step 3: Cancel and simplify further

Cancel \( x^3 \) from both the numerator and the denominator:\[ \frac{\sqrt{1 + \frac{4}{x^6}}}{-(1 - \frac{2}{x^3})} \]

Step 4: Evaluate the limit as \( x \to -\infty \)

As \( x \to -\infty \), the terms \( \frac{4}{x^6} \to 0 \) and \( \frac{2}{x^3} \to 0 \). Therefore, the expression becomes:\[ \frac{\sqrt{1}}{-(1)} = -1 \]

Key Concepts

Limit Evaluation

When approaching any limit problem, the key is understanding what happens as the variable of interest approaches a particular value, which in this case is as \( x \to -\infty \). Here, we are essentially testing the behavior of the function when \( x \) becomes very large negatively. To evaluate this, we want to simplify the expression, revealing the true behavior of the function at this extreme. Look for dominant terms, factor them out, and simplify to reveal the essence of the limit.
Before simplifying, checking the signs of expressions and final results is also crucial. Negative and positive infinities can result in different behaviors due to sign changes in leading terms. All these steps together help us correctly determine the limit's value or conclude its non-existence.

Infinite Limits

Infinite limits often appear when the variable approaches a very large positive or negative number, making expressions within either denominator or numerator very large or small. In these situations, terms with the highest powers of \( x \) take center stage, as they grow or shrink the most rapidly.
For our example:

• The numerator with \( \sqrt{1 + 4x^6} \) contains the term \( 4x^6 \).
• The denominator has \( 2 - x^3 \), with \( -x^3 \) leading.

Here, these leading terms dictate the behavior as \( x \to -\infty \). Therefore, understanding infinite limits involves isolating these powerful terms to determine how they affect the entire expression as \( x \) grows infinitely in the negative direction.

Dominant Terms Analysis

Dominant terms analysis is an invaluable technique for simplifying expressions when evaluating limits. By identifying which terms grow the fastest as the variable tends towards infinity or negative infinity, we can simplify the expression to focus on these major contributors without unnecessary complications. To apply dominant terms analysis:

• Identify the terms with the highest powers in both the numerator and the denominator.
• Divide every term by a chosen power of the variable, typically the one with the largest impact on the expression's growth.

In our exercise, we notice \( 4x^6 \) in the numerator and \( -x^3 \) in the denominator as dominant. Factoring and simplifying these terms allowed us to accurately calculate the limit. This method highlights how focusing on dominant behavior can lead to clearer, simpler solutions when considering complex rational expressions over infinite ranges.