Problem 17 - A Textbook of Differential Calculus

Question

\(\lim _{x \rightarrow a^{-}} \frac{\sqrt{x-b}-\sqrt{a-b}}{\left(x^{2}-a^{2}\right)},(a>b)\) (a) \(\frac{1}{4 a}\) (b) \(\frac{1}{a \sqrt{a-b}}\) (c) \(\frac{1}{2 a \sqrt{a-b}}\) (d) \(\frac{1}{4 a \sqrt{a-b}}\)

Step-by-Step Solution

Answer

The correct answer is (d): \(\frac{1}{4a\sqrt{a-b}}\).

Step 1: Identify the Type of Limit

The limit problem involves a difference of square roots in the numerator and a difference of squares in the denominator. This suggests the use of algebraic manipulation to simplify.

Step 2: Rationalize the Numerator

Multiply and divide the expression by the conjugate of the numerator: \(\frac{\sqrt{x-b}-\sqrt{a-b}}{x^2-a^2} \times \frac{\sqrt{x-b}+\sqrt{a-b}}{\sqrt{x-b}+\sqrt{a-b}}\). This gives: \[\frac{(\sqrt{x-b})^2-(\sqrt{a-b})^2}{(x^2-a^2)(\sqrt{x-b}+\sqrt{a-b})}\].

Step 3: Simplify the Numerator

Use the identity \((A-B)(A+B) = A^2 - B^2\). This simplifies the numerator: \((x-b) - (a-b) = x-a\).

Step 4: Simplify the Denominator

The expression is now \(\frac{x-a}{(x^2-a^2)(\sqrt{x-b}+\sqrt{a-b})}\). Factor \(x^2-a^2\) as \((x-a)(x+a)\).

Step 5: Cancel Common Factors

Cancel the factor \(x-a\) from numerator and denominator. The expression simplifies to \(\frac{1}{(x+a)(\sqrt{x-b}+\sqrt{a-b})}\).

Step 6: Evaluate the Limit as \(x \to a^-\)

Substitute \(x = a\) in the simplified expression: \[\frac{1}{(a+a)(\sqrt{a-b}+\sqrt{a-b})} = \frac{1}{2a\times2\sqrt{a-b}} = \frac{1}{4a\sqrt{a-b}}\].

Step 7: Choose the Correct Answer

The correct answer corresponds to option (d): \(\frac{1}{4a\sqrt{a-b}}\).

Key Concepts

Rationalization Technique

The rationalization technique is a special method in calculus used to simplify expressions involving roots, such as square roots, cube roots, etc. It is particularly useful in limit problems where the expression includes roots in the numerator or denominator. By rationalizing, we aim to remove the roots so that simplifications become easier and clearer.

In the original exercise, the expression in the numerator is \( \sqrt{x-b} - \sqrt{a-b} \), which involves a difference of square roots. To rationalize this expression, we multiply and divide by the conjugate, \( \sqrt{x-b} + \sqrt{a-b} \). The conjugate changes the expression into a differencing of squares, eliminating the roots:

• Multiply: \( (\sqrt{x-b})^2 - (\sqrt{a-b})^2 \)
• Simplify: \( (x-b) - (a-b) = x-a \)

Rationalization clears the way for further simplification, enabling us to cancel out common factors easily.

L'Hôpital's Rule

L'Hôpital's Rule is valuable when resolving indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Although not explicitly used in our example, it's crucial in similar problems where these forms appear after initial simplifications.

To apply L'Hôpital's Rule, you must first evaluate the function and confirm it results in an indeterminate form. If permissible, you then differentiate the numerator and the denominator separately and re-evaluate the limit using the resulting derivatives. Repeat this process until the indeterminate form resolves.

• Confirm the form: \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \)
• Differentiate numerator and denominator
• Evaluate or repeat if necessary

While this rule is powerful, understanding when and how to apply it comes with practice. In this exercise, careful simplification avoided the need for L'Hôpital's Rule.

Algebraic Simplification in Calculus

Algebraic simplification is a cornerstone in solving calculus problems, especially limits. It involves manipulating expressions to reveal simpler forms, making evaluation more straightforward. In our original problem, we see multiple steps of algebraic simplification to ease the calculation of the limit.

After rationalization, the initial complex fraction becomes manageable. By identifying common formulas such as \( (x^2 - a^2) = (x-a)(x+a) \) and simplifying accordingly:

• Numerator: From \( (x-b)-(a-b) = x-a \)
• Denominator: Factor \( x^2-a^2 \) as \( (x-a)(x+a) \)
• Cancel common terms: Remove \( x-a \) from both top and bottom

These steps reduce the expression considerably, ultimately simplifying the evaluation of the limit. Each algebraic action clears hurdles, ensuring a clear path to the solution. This approach highlights the elegance of calculus, where complex expressions become solvable through systematic simplification.