Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{2 x+1}}{x+4} $$

Step-by-Step Solution

Answer

The limit is \( \sqrt{2} \).

Step 1: Analyze the Expression

We need to find the limit of \( f(x) = \frac{\sqrt{2x+1}}{x+4} \) as \( x \to \infty \). To do this, we check how both the numerator and the denominator behave as \( x \) approaches infinity.

Step 2: Simplify the Expression

We simplify the expression \( \frac{\sqrt{2x+1}}{x+4} \). A common technique is to divide both the numerator and the denominator by the highest power of \( x \) in the denominator, which is \( x \).

Step 3: Simplify the Numerator and Denominator

The expression becomes:\[\frac{\sqrt{2x+1}}{x+4} = \frac{\sqrt{2x+1}}{x} \cdot \frac{x}{x+4} = \frac{\sqrt{2x+1}}{x} \cdot \frac{1}{1+\frac{4}{x}}\]Further simplify \( \frac{\sqrt{2x+1}}{x} \). We have:\[\frac{\sqrt{2x+1}}{x} = \frac{\sqrt{x^2(2 + \frac{1}{x})}}{x} = \frac{x\sqrt{2 + \frac{1}{x}}}{x} = \sqrt{2 + \frac{1}{x}}\]

Step 4: Evaluate the Limit

As \( x \to \infty \), \( \frac{1}{x} \to 0 \), so \( \sqrt{2 + \frac{1}{x}} \to \sqrt{2} \) and \( \frac{1}{1 + \frac{4}{x}} \to 1 \). Thus, the whole expression becomes:\[\sqrt{2 + \frac{1}{x}} \cdot \frac{1}{1+\frac{4}{x}} \to \sqrt{2} \cdot 1 = \sqrt{2}\]

Step 5: Conclusion

Finally, we conclude that the limit of the function as \( x \to \infty \) is \( \sqrt{2} \). Therefore, \(\lim_{x \to \infty} \frac{\sqrt{2x+1}}{x+4} = \sqrt{2}\).

Key Concepts

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input approaches a certain value, often infinity. In our exercise, we are interested in finding out what happens as the variable \( x \) becomes very large. This behavior helps us predict whether the function approaches a particular constant, becomes infinitely large, or tends to zero.

In the case of the expression \( \frac{\sqrt{2x+1}}{x+4} \), as \( x \) becomes exceedingly large (approaches infinity), the growth rates of the numerator and denominator determine the function's limit. The square root function in the numerator and the linear expression in the denominator each approach infinity, but at different rates. By analyzing their asymptotic behavior, we use mathematical techniques to see that this expression approaches the fixed value \( \sqrt{2} \).

Understanding asymptotic behavior is particularly important with rational functions and other expressions where both parts grow indefinitely. Asymptotic analysis thus provides insight into how closely the values of a function approach a limiting behavior or a horizontal asymptote.

Rational Functions

Rational functions are expressions made up of ratios of polynomials or algebraic expressions. Our exercise involves the rational function \( f(x) = \frac{\sqrt{2x+1}}{x+4} \). Here, the numerator \( \sqrt{2x+1} \) features a square root function, which while not a pure polynomial, still dictates the function's characteristics. The denominator, a simple linear polynomial \( x+4 \), completes the rational form.

Rational functions can have complex behaviors, showing different asymptotic tendencies as variables reach large or small values. Analyzing their limits involves determining how the top and bottom of the fraction contribute to the function's value as the inputs become large or small. In some cases, they might feature horizontal or vertical asymptotes or simply stabilize around a specific value.

In dealing with rational functions like ours, dividing terms by the highest power of \( x \) found in the denominator, as shown in the solution, simplifies them. This simplification allows you to see the core behavior and limits at infinity because the dominant terms dictate the outcome. Recognizing and manipulating these properties enables better understanding and problem-solving of rational function limits.

Limit Evaluation Techniques

Limit evaluation techniques are strategies used to compute the values that functions approach as the input grows indefinitely or nears a specific point. Several techniques exist to handle limits effectively, especially when dealing with complex expressions involving rational functions.

For the problem at hand, the technique used was dividing the numerator and the denominator by \( x \), the highest power of \( x \) in the denominator. This classic approach reduces the expression to its dominant terms, providing clearer insight into how parts of the expression contribute to the limit.

Let's break down the steps:

• Simplify both numerator and denominator by dividing through by the highest degree of \( x \) in the denominator.
• Identify which parts of the expression effectively become redundant as \( x \to \infty \), like \( \frac{4}{x} \to 0 \).
• Utilize properties of square roots and powers to further simplify the components.

By these processes, the expression \( \sqrt{2 + \frac{1}{x}} \cdot \frac{1}{1 + \frac{4}{x}} \) emerges, leading to the immediate evaluation of the limit as \( \sqrt{2} \).

Using these techniques is crucial, especially for students who are beginning to explore calculus, as they reveal intuitive ways to handle what initially appear to be complex mathematical puzzles.