Problem 29 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is \( \frac{a-b}{2} \).

Step 1: Identify the Expression

We are tasked with finding the limit as \( x \to \infty \) of \( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \). This type of limit often results in an indeterminate form, which requires algebraic manipulation to evaluate it.

Step 2: Rationalize the Expression

To eliminate the indeterminate form, multiply and divide the expression by its conjugate: \[ \lim_{x \to \infty} \frac{(\sqrt{x^2 + ax} - \sqrt{x^2 + bx})(\sqrt{x^2 + ax} + \sqrt{x^2 + bx})}{\sqrt{x^2 + ax} + \sqrt{x^2 + bx}} \]The numerator simplifies to:\[ (x^2 + ax) - (x^2 + bx) = ax - bx \] Thus, the expression becomes:\[ \lim_{x \to \infty} \frac{ax - bx}{\sqrt{x^2 + ax} + \sqrt{x^2 + bx}} \]

Step 3: Simplify the Expression

The expression simplifies to: \[ \lim_{x \to \infty} \frac{(a-b)x}{\sqrt{x^2 + ax} + \sqrt{x^2 + bx}} \]Factor out \( x \) from the square roots in the denominator:\[ \sqrt{x^2(1 + \frac{a}{x})} + \sqrt{x^2(1 + \frac{b}{x})} = x(\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}}) \]The expression is now:\[ \lim_{x \to \infty} \frac{(a-b)x}{x(\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}})} = \lim_{x \to \infty} \frac{a-b}{\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}}} \]

Step 4: Evaluate the Limit

As \( x \to \infty \), both \( \frac{a}{x} \) and \( \frac{b}{x} \) tend to 0. Therefore:\[ \lim_{x \to \infty} \sqrt{1 + \frac{a}{x}} = 1 \quad \text{and} \quad \lim_{x \to \infty} \sqrt{1 + \frac{b}{x}} = 1 \]Thus, the original limit becomes:\[ \lim_{x \to \infty} \frac{a-b}{1 + 1} = \frac{a-b}{2} \]

Step 5: Conclusion

The limit can be determined and it simplifies to \( \frac{a-b}{2} \). Therefore, the limit of the expression as \( x \to \infty \) is \( \frac{a-b}{2} \).

Key Concepts

Indeterminate Forms

When evaluating limits, especially as expressions tend toward infinity, indeterminate forms often surface. These forms are expressions where substitution results in ambiguous outcomes like \( \frac{0}{0} \) or \( \infty - \infty \). In our exercise, \( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \) initially appears to be indeterminate as \( x \to \infty \).

Indeterminate forms signal complexity in calculations requiring further manipulation for resolution. By employing techniques like algebraic rearrangement or multiplying by conjugates, these forms are often transformed into solvable expressions. Recognizing and handling indeterminate forms is crucial in finding limits, helping us to determine precise behavior of functions in calculus.

Conjugate Method

The conjugate method is a powerful technique used to simplify expressions, especially in the presence of square roots. It involves multiplying and dividing the original expression by its conjugate. The expression \( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \) is paired with its conjugate \( \sqrt{x^2 + ax} + \sqrt{x^2 + bx} \).

Applying the conjugate method transforms the difference of two roots into a rational expression. By multiplying and dividing by the conjugate, we utilize the difference of squares: \((a - b)(a + b) = a^2 - b^2\). This strategy simplifies the expression, removing roots and revealing a path to find the limit. Using conjugates is especially useful when dealing with indeterminate forms like \( \infty - \infty \).

Rationalizing the Numerator

Rationalizing is the process of eliminating square roots or other irrational numbers from the numerator or denominator. In our example, this is achieved by multiplying by the conjugate to remove indeterminate forms.

By rationalizing \( \sqrt{x^2 + ax} - \sqrt{x^2 + bx} \), we transform the numerator into \( ax - bx = (a-b)x \). As a result, the expression becomes more manageable, allowing for straightforward division and making the limit easier to evaluate. Rationalization is invaluable in calculus, making complex expressions that include radicals more tractable.

Asymptotic Behavior

Analyzing asymptotic behavior involves understanding how functions behave as they approach infinite limits. In this exercise, the expression's behavior as \( x \to \infty \) provides insights into its long-term tendencies.

Factoring \( x \) out of the radicals highlights this behavior, yielding \( x(\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}}) \). As \( x \) increases, terms \( \frac{a}{x} \) and \( \frac{b}{x} \) approach zero, simplifying the radicals to their limits at \( 1 \). This insight transforms the challenge into finding the behavior of \( \frac{a-b}{2} \) as \( x \to \infty \).

Asymptotic analysis in calculus helps in predicting function behavior at extremes, which is crucial for understanding limits and continuity.