Problem 36 - Calculus Early Transcendentals

Question

Use the definition of a limit to prove the following results. \(\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}=4\) (Hint: Multiply the numerator and denominator by \(\sqrt{x}+2 .)\)

Step-by-Step Solution

Answer

Question: Prove that \(\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}=4\) using the given hint to multiply the numerator and denominator by \(\sqrt{x} + 2\). Answer: We multiplied the function by its conjugate, simplified the expression to \(\sqrt{x}+2\), and applied the limit as \(x \rightarrow 4\) to arrive at the answer, 4.

Step 1: Multiply by the conjugate

First, we will multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{x} + 2\). This is done to simplify the expression. \((\frac{x-4}{\sqrt{x}-2})(\frac{\sqrt{x}+2}{\sqrt{x}+2}) = \frac{(x-4)(\sqrt{x}+2)}{(\sqrt{x}-2)(\sqrt{x}+2)}\)

Step 2: Simplify the expression

Now, we can simplify the expression by applying the difference of squares formula in the denominator and distributing in the numerator: \(\frac{(x-4)(\sqrt{x}+2)}{(\sqrt{x}-2)(\sqrt{x}+2)} = \frac{(x-4)(\sqrt{x}+2)}{x-4} = \sqrt{x} + 2\)

Step 3: Apply the limit

Now that we have a simplified expression, we can apply the limit as \(x \rightarrow 4\): \(\lim _{x \rightarrow 4} (\sqrt{x} + 2) = \sqrt{4} + 2 = 2 + 2 = 4\) So, we have proved that \(\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}=4\).

Key Concepts

Conjugate Multiplication

Conjugate multiplication is a handy technique used to simplify expressions, especially those involving roots or radicals. If you encounter an expression with a radical in the denominator, multiplying by the conjugate can help you eliminate it. The conjugate of a binomial expression like \(a - b\) is \(a + b\), and vice versa. This process revolves around the principle that multiplying conjugates results in a difference of squares, effectively removing the radical. In the given exercise, we had a denominator \(\sqrt{x} - 2\). By multiplying both the numerator and the denominator by the conjugate, \(\sqrt{x} + 2\), we are preparing the expression for further simplification without changing its value. This step ensures we are not dividing by zero or introducing any extraneous terms.

Difference of Squares

The difference of squares is a formula that comes into play after multiplying an expression by its conjugate. It states that \((a-b)(a+b) = a^2 - b^2\). When applying the conjugate multiplication, the denominator becomes a perfect candidate for using this formula. For the expression \((\sqrt{x} - 2)(\sqrt{x} + 2)\), using the difference of squares simplifies the denominator to one single term: \(x - 4\). This form is much easier to handle and removes the radical, allowing further simplification of the overall fraction. Recognizing your ability to switch from a complex expression to an algebraically cleaner one like this is a powerful tool when solving limits and simplifying math problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by using known algebraic rules and properties. In this exercise, after applying the conjugate multiplication and using the difference of squares, we were left with a fraction \(\frac{(x-4)(\sqrt{x}+2)}{x-4}\). Here, the \(x-4\) terms in the numerator and the denominator cancel each other out, assuming \(x eq 4\) to avoid division by zero. This leaves us with \(\sqrt{x} + 2\). By simplifying, we eliminate complex fractions or terms and make evaluating processes like limits straightforward. This is crucial for proceeding confidently with solving limits, as a simpler expression reduces errors and makes substitution easier.