Problem 50 - Calculus Early Transcendentals

Question

Evaluate the following limits, where a and \(b\) are fixed real numbers. \(\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{\sqrt{x}-\sqrt{a}}, a>0\)

Step-by-Step Solution

Answer

Expression: $$\frac{x^2-a^2}{\sqrt{x}-\sqrt{a}}$$ Answer: The limit of the given expression as x approaches a is \(4a\sqrt{a}\).

Step 1: Factor the numerator

The numerator can be factored using the difference of squares formula. \((x^2-a^2) = (x-a)(x+a)\). The expression becomes: $$\frac{(x-a)(x+a)}{\sqrt{x}-\sqrt{a}}$$

Step 2: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x}+\sqrt{a}\) $$\frac{(x-a)(x+a)}{\sqrt{x}-\sqrt{a}} \cdot \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}} = \frac{(x-a)(x+a)(\sqrt{x}+\sqrt{a})}{x-a}$$

Step 3: Simplify the expression

The \((x-a)\) terms in the numerator and denominator cancel each other out: $$\frac{(x+a)(\sqrt{x}+\sqrt{a})}{1}$$

Step 4: Evaluate the limit

Now, we can find the limit as x approaches a: $$\lim _{x \rightarrow a} (x+a)(\sqrt{x}+\sqrt{a})$$ By substituting x by a, we get: $$\lim _{x \rightarrow a} (a+a)(\sqrt{a}+\sqrt{a}) = 2a \cdot 2\sqrt{a} = 4a\sqrt{a}$$ The limit of the given expression as x approaches a is \(4a\sqrt{a}\).

Key Concepts

Difference of Squares

The difference of squares is a handy algebraic formula used to simplify expressions involving squared terms. It follows the pattern

• \(x^2 - a^2 = (x-a)(x+a)\)

This formula helps in breaking down a complex expression into simpler factors. When you see an expression where two square terms are subtracted, you can immediately apply this technique. For example, in our problem, the numerator \(x^2 - a^2\) simplifies to \((x-a)(x+a)\). This step is crucial before proceeding to further manipulations. Such factorization enables canceling common terms and simplifies subsequent operations, making it easier to evaluate limits or solve equations.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any square roots or irrational numbers from the denominator of a fraction. To do this, we use the conjugate. If the denominator has the form

• \(\sqrt{x} - \sqrt{a}\)

then its conjugate will be

• \(\sqrt{x} + \sqrt{a}\)

Multiplying both the numerator and the denominator by this conjugate helps to simplify the expression by removing the irrational term. Here, after multiplying by the conjugate, the term \((\sqrt{x})^2 - (\sqrt{a})^2\) simplifies to \(x - a\). This allows further simplification, making it easier to evaluate the limit. It helps to see the rationalization effect as transforming the expression into a form that's "limit-ready."

Limits and Continuity

Limits are fundamental in calculus, providing a way to understand the behavior of functions as inputs approach specific values. Continuity, a related concept, ensures that functions behave predictably without abrupt changes. To find a limit as \(x\) approaches a value, like in our exercise where \(x \to a\), requires simplifying the expression. Once simplified, if the function is continuous at the point, you can directly substitute the \(x\) value. In our solved example, the simplification process reveals a continuous function after canceling terms and rationalizing the denominator. This reveals the ultimate value of the limit. It's this predictability in behavior, algebraic manipulation, and simplification which are key to evaluating limits.

Algebraic Manipulation

Algebraic manipulation involves using various techniques to rearrange and simplify expressions for easier computation. It often includes factoring, expanding, simplifying fractions, and canceling terms. In our solution, algebraic manipulation involved several key steps. First, we factored the numerator using the difference of squares. Then, we rationalized the denominator using conjugate multiplication. Finally, by canceling common terms, we simplified the expression. Such maneuvers transform a complex-looking problem into a straightforward one. By mastering these skills, limits and other calculus operations become more accessible, allowing for direct substitution and evaluation of limits.