Question

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

Short answer

Answer: The limit of the function is $3\sqrt{a}$.

Step-by-step solution

Identify the Indeterminate Form

We see that as x approaches 'a', both the numerator and the denominator go to 0. Therefore, we have the indeterminate form 0/0, which requires further simplification.

Rationalize the Denominator

In order to simplify the expression, we can multiply the numerator and the denominator by the conjugate of the denominator (√x + √a): $$\lim _{x \rightarrow a} \frac{(x-a)(\sqrt{x}+\sqrt{a})}{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})}$$

Simplify the Expression

Next, we need to expand the numerator and the denominator, and then cancel out any common terms (if possible). Numerator: \((x-a)(\sqrt{x}+\sqrt{a}) = x\sqrt{x} - a\sqrt{x} + x\sqrt{a} - a\sqrt{a}\) Denominator: \((\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a}) = (\sqrt{x})^2 - (\sqrt{a})^2 = x - a\) Now, we have: $$\lim _{x \rightarrow a} \frac{x\sqrt{x} - a\sqrt{x} + x\sqrt{a} - a\sqrt{a}}{x-a}$$

Cancel Common Factors

We can factor out an (x-a) in the numerator and then cancel it with the denominator: $$\lim _{x \rightarrow a} \frac{(x-a)(\sqrt{x}+\sqrt{a}+\sqrt{a})}{x-a}$$ After canceling the common factors (x-a), we get: $$\lim _{x \rightarrow a} (\sqrt{x}+\sqrt{a}+\sqrt{a})$$

Evaluate the Limit

After simplifying, we can now substitute 'a' for 'x' and find the limit: $$\lim _{x \rightarrow a} (\sqrt{x}+\sqrt{a}+\sqrt{a}) = \sqrt{a}+\sqrt{a}+\sqrt{a} = 3\sqrt{a}$$ So, the limit of the given function is: $$\lim _{x \rightarrow a} \frac{x-a}{\sqrt{x}-\sqrt{a}} = 3\sqrt{a}$$

Key concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used to simplify expressions that contain roots in their denominators. When finding limits like in our example, rationalizing can help us eliminate these roots, allowing for easier manipulation of the expression.

For example, consider the expression \(\frac{x-a}{\sqrt{x}-\sqrt{a}}\). Here, the denominator contains square roots. To rationalize it, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x}+\sqrt{a}\).

This step transforms our expression into:\[\lim _{x \rightarrow a} \frac{(x-a)(\sqrt{x}+\sqrt{a})}{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})}\] The beauty of this transformation is that the denominator becomes a difference of squares, eliminating the roots.

Indeterminate Form

In limit problems, especially ones involving zeroes in the numerator and denominator, we often face the indeterminate form \(\frac{0}{0}\). This form occurs in our problem when \(x \rightarrow a\), as both the top and bottom of the fraction equal zero.

Indeterminate forms tell us that direct substitution will not work to find the limit. They require further simplification techniques, such as rationalization, to move beyond the indeterminate state.

Understanding that an expression is in an indeterminate form is the first step in determining that more work is necessary to evaluate the limit.

Limit Simplification

Simplifying limits involves transforming an original complex expression into a simpler one that is easier to evaluate. In our problem, after multiplying by the conjugate, the expression simplifies significantly as common terms can be canceled.

We start with the expanded form from rationalization:

• Numerator: \(x\sqrt{x} - a\sqrt{x} + x\sqrt{a} - a\sqrt{a}\)
• Denominator: \(x-a\)

To simplify further, we notice that the term \((x-a)\) emerges naturally in the numerator, allowing it to be canceled with the denominator. This results in no indeterminate form and yields a clean expression for limit evaluation.

Conjugate Multiplication

Conjugate multiplication is a technique used in algebra to simplify expressions containing roots. We multiply the expression by a conjugate, which in this context means changing the sign between two terms in a binomial.

For example, the conjugate of \(\sqrt{x}-\sqrt{a}\) is \(\sqrt{x}+\sqrt{a}\). Conjugate multiplication helps create a difference of squares, which eliminates root expressions after multiplication.

Applying conjugates is crucial because it transforms complex root expressions, making them easier to simplify. Once transformed, terms can often be factored and canceled, streamlining the process of limit evaluation.