Problem 80 - Thomas Calculus

Question

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}(\sqrt{x+9}-\sqrt{x+4})$$

Step-by-Step Solution

Answer

The limit is 0 as \( x \) approaches infinity.

Step 1: Understand the Problem

We need to evaluate the limit \( \lim_{x \to \infty}(\sqrt{x+9} - \sqrt{x+4}). \) The expression involves square roots and a difference of terms as \( x \) approaches infinity.

Step 2: Rationalize the Expression

To simplify the limit, multiply the expression by the conjugate over itself: \( \frac{\sqrt{x+9} - \sqrt{x+4}}{1} \cdot \frac{\sqrt{x+9} + \sqrt{x+4}}{\sqrt{x+9} + \sqrt{x+4}}.\)

Step 3: Simplify the Numerator

Multiply the expression inside the limit: \( \lim_{x \to \infty} \frac{(\sqrt{x+9} - \sqrt{x+4})(\sqrt{x+9} + \sqrt{x+4})}{\sqrt{x+9} + \sqrt{x+4}}.\) The numerator becomes \((x+9) - (x+4) = 5.\)

Step 4: Simplify the Denominator

The limit now is \( \lim_{x \to \infty} \frac{5}{\sqrt{x+9} + \sqrt{x+4}}.\) To simplify further, factor \( x \) from each square root, we have \( \sqrt{x(1 + 9/x)} + \sqrt{x(1 + 4/x)} \).

Step 5: Evaluate the Limit

As \( x \to \infty,\) \( \sqrt{1 + 9/x} \to 1 \) and \( \sqrt{1 + 4/x} \to 1.\) Therefore, the limit reduces to \( \lim_{x \to \infty} \frac{5}{\sqrt{x}(\sqrt{1 + 9/x} + \sqrt{1 + 4/x})} = 0. \) This is because \( \frac{5}{\sqrt{x}(2)} \to 0 \) as \( x \) approaches infinity.

Key Concepts

Rationalization technique

The rationalization technique is a mathematical tool used to simplify expressions, often involving square roots. It's particularly useful when dealing with limits where direct substitution leads to indeterminacy. By rationalizing, you replace a square root term with its conjugate, helping eliminate these roots and simplifying the computation.In our given problem, we start with \( \lim_{x \to \infty}(\sqrt{x+9} - \sqrt{x+4}) \). The direct approach of trying to substitute \( \infty \) will not work directly due to the roots. Here is where rationalization comes in handy:

• Multiply the expression by the conjugate of the numerator: \((\sqrt{x+9} + \sqrt{x+4})\).
• The purpose is to transform \( (\sqrt{x+9} - \sqrt{x+4})(\sqrt{x+9} + \sqrt{x+4}) \) into a square difference.
• By multiplying these, you obtain \((x+9) - (x+4) = 5\), dramatically simplifying the expression.

This technique clears the roots from the numerator, allowing the problem to simplify to a more manageable form, which in turn makes it easier to evaluate the limit.

Infinity limit

One key aspect of solving limits is understanding how they behave as they approach infinity. A limit at infinity explores the behavior of functions as the variable grows indefinitely large. This could often give us insight into the long-term behavior or 'end' behavior of the function.In this exercise, we're examining \( \lim_{x \to \infty} \frac{5}{\sqrt{x+9} + \sqrt{x+4}} \). This means we're interested in knowing what happens to the fraction as \( x \) tends toward infinity:

• As \( x \) becomes very large, the expressions \( \sqrt{x+9} \) and \( \sqrt{x+4} \) both closely resemble \( \sqrt{x} \).
• Hence, the denominator can be approximated as \( \sqrt{x}(\sqrt{1 + 9/x} + \sqrt{1 + 4/x}) \).
• Both terms \( \sqrt{1 + 9/x} \) and \( \sqrt{1 + 4/x} \) tend toward 1, simplifying the entire denominator to resemble \( \sqrt{x}(2) \).

Thus, the entire limit becomes \( \frac{5}{\sqrt{x} \cdot 2} \), which approaches 0 as \( x \to \infty \). This approach of examining the behavior at infinity is crucial in understanding the essence of how functions change over large inputs.

Simplifying square roots

Simplifying square roots is fundamental when dealing with expressions involving limits and irrational numbers. It's a process aimed at transforming complicated root expressions into a more recognizable and workable form.During the calculation of our limit, we come across terms like \( \sqrt{x+9} \) and \( \sqrt{x+4} \), which at first glance seem difficult to handle, especially as we consider infinity. Here's how simplifying works:

• Rewrite \( \sqrt{x+9} \) using factoring within the root: \( \sqrt{x(1 + 9/x)} \).
• Similarly, \( \sqrt{x+4} \) becomes \( \sqrt{x(1 + 4/x)} \).
• As \( x \to \infty \), both \( 9/x \) and \( 4/x \) trend towards 0. Hence, \( \sqrt{1 + 9/x} \) and \( \sqrt{1 + 4/x} \) each simplify to approximately 1.

This simplification helps us tackle the limit expression more effectively by reducing complexity and focusing only on the dominant terms. Understanding this technique is invaluable for reliably simplifying complex square roots found in many mathematical contexts.