Problem 28 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

We need to find the limit of the expression \( \sqrt{4x^2 + 3x} + 2x \) as \( x \) approaches \( -\infty \). This means approaching very large negative values for \( x \) and determining what happens to the expression as a whole.

Step 2: Simplify the Square Root

Rewrite the square root term \( \sqrt{4x^2 + 3x} \). Notice that as \( x \to -\infty \), \( 4x^2 \) is dominant compared to \( 3x \), so \( \sqrt{4x^2 + 3x} \) behaves like \( \sqrt{4x^2} = 2|x| \). Since \( x \to -\infty \), \( |x| = -x \), thus \( 2|x| = -2x \).

Step 3: Analyze the Expression

Substitute the simplified term into the original expression: \( \sqrt{4x^2 + 3x} + 2x \approx -2x + 2x \). This simplifies to zero, but let's consider the error term to refine our result.

Step 4: Determine Error Term

The error in replacing \( \sqrt{4x^2 + 3x} \) by \( -2x \) is \( \sqrt{4x^2 + 3x} - (-2x) = \sqrt{4x^2 + 3x} + 2x \). To find this difference as \( x \to -\infty \), factor further: \( \sqrt{4x^2 + 3x} = \sqrt{4x^2(1 + \frac{3}{4x})} \approx 2x\sqrt{1 + \frac{3}{4x^2}} \approx 2x(1 + \frac{3}{8x}) \approx 2x + \frac{3}{4x} \).

Step 5: Calculate the Limit of the Error Term

Consider \( \frac{3}{4x} \). As \( x \to -\infty \), \( \frac{1}{x} \to 0 \), thus \( \frac{3}{4x} \to 0 \). Hence, the limit of the error term is negligible.

Step 6: Conclude the Limit

Adding up the dominant part and the error term, we find that \( \lim_{x \to -\infty} \left( \sqrt{4x^2 + 3x} + 2x \right) = \lim_{x \to -\infty} \left( -\frac{3}{4x} \right) = 0 \). Therefore, the limit is 0.

Key Concepts

Asymptotic behavior

Asymptotic behavior in calculus refers to how a function behaves as the input values become very large or very small. In this problem, we are observing the behavior of the function \( \sqrt{4x^2 + 3x} + 2x \) as \( x \) approaches negative infinity. Understanding asymptotic behavior is essential because it helps simplify complex expressions by focusing on the terms that have the most significant impact when \( x \) is extremely large or small.In the given expression, as \( x \to -\infty \), the largest term in \( 4x^2 + 3x \) is \( 4x^2 \). This is because polynomial terms with higher powers grow faster than those with lower powers when \( x \) becomes very large or very negative. As a result, the minor terms become less significant in the context of larger terms.By examining asymptotic behavior, we can focus on simplifying the dominant parts of the expression, which guides us toward resolving the behavior of the entire function. This approach makes complex limits much easier to evaluate.

Square root simplification

Simplifying square roots is crucial when evaluating limits, especially when working with terms that include polynomials. In this problem, the expression \( \sqrt{4x^2 + 3x} \) is initially quite complex due to the combination of terms underneath the square root.To simplify, notice that \( 4x^2 \) is the dominant term as \( x \to -\infty \). Since its contribution is significantly larger than the linear term \( 3x \), we can approximate \( \sqrt{4x^2 + 3x} \) as \( \sqrt{4x^2} \). This simplifies to \( 2|x| \), but as \( x \) approaches negative infinity, \( |x| = -x \). Therefore, \( \sqrt{4x^2 + 3x} \approx -2x \).This simplification is an essential step because reducing the complexity of the square root term allows easier calculation of the limit. It also helps in identifying any error terms that might arise, which we deal with separately. Ensuring each part is as simple as possible is key in finding the correct limit.

Dominant term analysis

Analyzing dominant terms is a strategy used in calculus to handle expressions where some terms significantly outweigh others as the variable approaches infinity or negative infinity.In the expression \( \sqrt{4x^2 + 3x} + 2x \), we looked at the square root first to identify the dominant term in \( 4x^2 + 3x \), which is \( 4x^2 \). Simplifying \( \sqrt{4x^2} \) gives us a sense of the scale of \( \sqrt{4x^2 + 3x} \) compared to the entire expression.Being able to recognize dominant terms allows us to simplify complicated evaluations into manageable computations. The asymptotic effect of different parts of an expression shows how each contributes to the result. By replacing the root with its dominant behavior, and then dealing with the leftover terms as error or negligible parts, we streamline the path towards concluding the overall limit.