Problem 45 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow a} \frac{x-a}{\sqrt{x}-\sqrt{a}}, a>0$$

Step-by-Step Solution

Answer

Answer: The limit of the expression is \(2\sqrt{a}\).

Step 1: Identify the given expression and the limit

The expression is given by: $$f(x) = \frac{x-a}{\sqrt{x}-\sqrt{a}}$$ We want to find the limit of this expression as \(x \rightarrow a\).

Step 2: Rationalize the denominator

In order to rationalize the denominator, we'll multiply both numerator and denominator of the fraction by the conjugate of the denominator, which is \(\sqrt{x} + \sqrt{a}\): $$f(x) = \frac{(x-a)(\sqrt{x}+\sqrt{a})}{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})}$$

Step 3: Simplify the expression

Now, we'll simplify both the numerator and denominator: Numerator: \((x-a)(\sqrt{x}+\sqrt{a}) = x\sqrt{x} + x\sqrt{a} - a\sqrt{x} - a\sqrt{a}\) Denominator: \((\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a}) = (\sqrt{x})^2 - (\sqrt{a})^2 = x - a\) Now we have the simplified expression: $$f(x) = \frac{x\sqrt{x} + x\sqrt{a} - a\sqrt{x} - a\sqrt{a}}{x - a}$$

Step 4: Find the limit as x approaches a

Now we can find the limit of the simplified expression as \(x \rightarrow a\): $$\lim_{x \rightarrow a} \frac{x\sqrt{x} + x\sqrt{a} - a\sqrt{x} - a\sqrt{a}}{x - a}$$ The first thing to notice is that the expression is of the form \(\frac{0}{0}\) when we substitute \(x=a\). This suggests that we can perform some algebraic simplification before finding the limit. Factoring out an \(x - a\) term from the numerator: $$f(x) = \frac{(x - a)(\sqrt{x} + \sqrt{a})}{x - a}$$ Cancel out the \((x - a)\) term in the numerator and denominator: $$f(x) = \sqrt{x} + \sqrt{a}$$ Finally, we can find the limit easily: $$\lim_{x \rightarrow a} \sqrt{x} + \sqrt{a} = \sqrt{a} + \sqrt{a} = \boxed{2\sqrt{a}}$$

Key Concepts

Rationalization Technique

In the context of limits, the rationalization technique is often used to simplify expressions that have radicals or square roots in the numerator or denominator. It helps to eliminate the radical by multiplying the numerator and the denominator of a fraction by the conjugate of the radical part. This technique can be extremely useful when dealing with limits because it transforms a complex-looking expression into a more manageable form. For instance, if we have a denominator like \( \sqrt{x} - \sqrt{a} \), multiplying by its conjugate \( \sqrt{x} + \sqrt{a} \) clears the radical, leaving us with a simpler expression. After rationalization, it becomes easier to find the limit by algebraic simplification. Remember, the conjugate involves changing the sign between the radical terms, assisting in canceling out terms, especially when the expression appears as an indeterminate form.

Algebraic Simplification

Algebraic simplification involves reducing an expression into its simplest form, often by performing arithmetic operations or factoring. In this case, after rationalizing the denominator, the expression \( \frac{(x-a)(\sqrt{x}+\sqrt{a})}{x-a} \) emerges. Simplifying this includes canceling out common factors in the numerator and denominator, such as \( x-a \), which removes the \( 0/0 \) indeterminacy in the expression. Simplification transforms our algebraic expression into a form where the limit can easily be evaluated. The goal of simplification in limits is to make closer analysis possible, especially as the variable approaches a particular value. By doing so, students can see through the complexity to determine how expressions behave as they approach a critical point. It is a foundational skill that requires practice and an understanding of algebraic identities and properties.

Indeterminate Forms

Indeterminate forms are expressions that arise in calculus when evaluating limits and provide no immediate information about the limit itself. The most common indeterminate forms are \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), and \( 0 \times \infty \), among others. In this exercise, initially substituting \( x = a \) into the function \( f(x) = \frac{x-a}{\sqrt{x}-\sqrt{a}} \) results in the form \( \frac{0}{0} \). This signals the need for additional algebraic manipulation, such as rationalization or simplification, rather than attempting a straightforward limit substitution. Dealing with these forms requires recognizing the expression type and applying calculus techniques like L'Hôpital's Rule or algebraic simplification to ascertain the limit. Understanding that such forms exist is crucial—they tell us there is further work to be done rather than concluding that a limit does not exist.