Question

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

Short answer

Question: Evaluate the limit of the function \(\frac{x^2 - a^2}{\sqrt{x} - \sqrt{a}}\) as \(x\) approaches \(a\), where \(a > 0\). Answer: The limit of the function as \(x\) approaches \(a\) is \(4a\sqrt{a}\), where \(a > 0\).

Step-by-step solution

Factor the difference of squares

First, we will factor the numerator expression, which is a difference of squares. Recall the difference of squares formula: \((p^2 - q^2) = (p + q)(p - q)\). Applying this formula to \((x^2 - a^2)\), we get: \[ x^2 - a^2 = (x + a)(x - a) \]

Rationalize the denominator

Next, we will rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate is the same expression with the sign between the terms reversed. In our case, the conjugate of \((\sqrt{x} - \sqrt{a})\) is \((\sqrt{x} + \sqrt{a})\). Thus, we multiply our function by \(\frac{(\sqrt{x} + \sqrt{a})}{(\sqrt{x} + \sqrt{a})}\). We get: \[ \frac{(x^2 - a^2)(\sqrt{x} + \sqrt{a})}{(\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a})} \]

Simplify expressions

Now, we will replace the factored form of the numerator \((x^2 - a^2)\) with \((x + a)(x - a)\), and expand the numerator. We will also expand the denominator using the conjugate formula. Thus, we have: \[ \frac{[(x + a)(x - a)](\sqrt{x} + \sqrt{a})}{(\sqrt{x} - \sqrt{a})(\sqrt{x} + \sqrt{a})} \] Expanding the denominator using conjugate formula, we get: \[ \frac{[(x + a)(x - a)](\sqrt{x} + \sqrt{a})}{x - a} \]

Simplify and cancel factors

Observe that the expression \((x-a)\) is a common factor in both the numerator and the denominator. We cancel this common factor, and we get: \[ \frac{(x + a)(\sqrt{x} + \sqrt{a})}{1} \] which simplifies to: \[ (x + a)(\sqrt{x} + \sqrt{a}) \]

Evaluate the limit

Now that we have simplified the function, we can apply the limit as \(x\) approaches \(a\). By simply substituting \(a\) in the expression, we get: \[ \lim_{x \rightarrow a} (x + a)(\sqrt{x} + \sqrt{a}) = (a + a)(\sqrt{a} + \sqrt{a}) \] Finally, we get: \[ 2a(2 \sqrt{a}) = 4a\sqrt{a} \] The limit of the given function as \(x\) approaches \(a\) is \(4a\sqrt{a}\), where \(a > 0\).

Key concepts

Factoring Polynomials

Factoring polynomials is a way of breaking down expressions into simpler components called factors. This process makes complex problems easier to handle. In our exercise, we are dealing with a specific type of polynomial factorization known as the "difference of squares."

Why is it called "difference of squares"? Because the expression \(x^2 - a^2\) represents one square number subtracted from another. This can be rewritten using the formula:

• Difference of squares formula: \(p^2 - q^2 = (p + q)(p - q)\)

By applying this formula to \(x^2 - a^2\), it factors into \((x + a)(x - a)\). This step is crucial before moving on to simplify the limit expression. By expressing it in terms of its factors, we can easily cancel out and simplify terms in later steps. Understanding this factorization allows us to simplify polynomial expressions efficiently and is an essential skill for evaluating limits, solving equations, and more.

Difference of Squares

The difference of squares is a useful algebraic identity that simplifies expressions involving subtraction of two perfect squares like \(x^2 - a^2\). This identity is expressed by \(u^2 - v^2 = (u + v)(u - v)\). It transforms the difference of two square terms into a product of two binomials.

This identity is particularly useful in limit problems, where simplification is needed to evaluate indeterminate forms. In our exercise, applying this identity to the polynomial \(x^2 - a^2\) simplifies it into a product, making it easier to manage when combined with the denominator.

• Eliminates indeterminate forms
• Facilitates the cancellation of common terms

Once the factors are laid out, it often reveals common factors that simplify the calculation, especially when algebraic manipulation is needed to cancel out terms causing indeterminate results. Recognizing and correctly applying this formula is key to mastering algebraic simplification techniques.

Rationalizing the Denominator

Rationalizing the denominator is a method used to eliminate radicals, like square roots, from the denominator of a fraction. It simplifies expressions by making them easier to work with. In problems involving limits, this technique is often applied to resolve indeterminate forms.

In our exercise, the original denominator contains a term \(\sqrt{x} - \sqrt{a}\), making simplification challenging. To rationalize, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x} + \sqrt{a}\). This uses the identity:

• Product of conjugates: \( (p - q)(p + q) = p^2 - q^2\)

This action transforms the denominator into a simple difference of squares, \(x - a\), effectively removing the square root without altering the expression’s value. Rationalizing the denominator ensures that we achieve a simpler form of the limit expression, allowing us to straightforwardly evaluate or further simplify. This technique is essential for dealing with radicands in algebra and calculus, particularly when evaluating limits.