Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow \infty} \frac{2 x+1}{\sqrt{x^{2}+3}}\). Hint: Divide numerator and denominator \(x\). Note that, for \(x>0, \sqrt{x^{2}+3} / x=\sqrt{\left(x^{2}+3\right) / x^{2}}\). py

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Rewrite the Limit Expression

Exponentially, as x approaches infinity, we rewrite the original expression:\[ \lim _{x \rightarrow \infty} \frac{2x+1}{\sqrt{x^2+3}} \]The hint suggests dividing the numerator and the denominator by \(x\) to simplify this expression.

Step 2: Simplify the Numerator

Divide the entire numerator, \(2x+1\), by \(x\):\[ \frac{2x+1}{x} = \frac{2x}{x} + \frac{1}{x} = 2 + \frac{1}{x} \]

Step 3: Simplify the Denominator

Apply the mathematical operation suggested in the hint for the denominator:\[ \frac{\sqrt{x^2+3}}{x} = \sqrt{\frac{x^2+3}{x^2}} \]This simplifies further to:\[ \sqrt{\frac{x^2}{x^2} + \frac{3}{x^2}} = \sqrt{1 + \frac{3}{x^2}} \]

Step 4: Rewrite the Expression in Simplified Form

Substitute the simplified expressions from Steps 2 and 3 into the limit expression:\[ \lim_{x \rightarrow \infty} \frac{2 + \frac{1}{x}}{\sqrt{1 + \frac{3}{x^2}}} \]

Step 5: Evaluate the Limit as x Approaches Infinity

As \(x\) approaches infinity, \(\frac{1}{x}\) approaches 0, and \(\frac{3}{x^2}\) also approaches 0. This changes the expression to:\[ \lim_{x \rightarrow \infty} \frac{2 + 0}{\sqrt{1 + 0}} = \frac{2}{1} = 2 \]

Key Concepts

Infinite Limits

When we talk about limits as a variable approaches infinity, we are considering the behavior of a function or expression as the input becomes extremely large or small. In our exercise, we are interested in what happens to the expression \( \lim _{x \rightarrow \infty} \frac{2x+1}{\sqrt{x^2+3}} \). Our task is to figure out what this fraction tends to as \( x \) gets larger and larger.

Infinite limits deal with cases where either the numerator or denominator of a fraction grows without bound. However, what matters is how fast each part is growing.

• If the numerator grows faster, the overall expression tends towards infinity.
• If the denominator grows faster, the expression tends towards zero.
• If they grow at the same rate, as in our exercise, it necessitates further simplification to find the limit.

Limit Simplification

To simplify limits, especially those involving infinite terms, we often divide terms to simplify the expression. In this problem, we are advised to divide by \( x \) in both the numerator and the denominator. This helps us manage the large terms more effectively.

Here's how it works:

• When dividing the numerator \( 2x+1 \) by \( x \), we get \( 2 + \frac{1}{x} \).
• Similarly, by dividing the denominator \( \sqrt{x^2+3} \) by \( x \), we transform it into \( \sqrt{1+\frac{3}{x^2}} \).

This approach simplifies the expression to \( \lim_{x \rightarrow \infty} \frac{2 + \frac{1}{x}}{\sqrt{1 + \frac{3}{x^2}}} \), making it easier to evaluate as \( x \) heads towards infinity.

Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P \) and \( Q \) are polynomials. In our exercise, \( \frac{2x+1}{\sqrt{x^2+3}} \) is a rational function involving a polynomial in the numerator and a square root function in the denominator.

Handling rational functions involves looking at the degrees of the polynomials:

• If the degree of the polynomial in the numerator is higher, the limit as \( x \) approaches infinity is typically infinity.
• If the degree in the denominator is higher, the limit is typically zero.
• If they are the same, the limit depends on the coefficients of the highest degree terms.

In our case, simplifying the function shows both the numerator and denominator grow at the same rate, leading to the need for further simplification to find that the limit equals 2.

Square Root Limits

Square root limits involve expressions where one or more terms within a limit are under a square root, often making them appear complex initially. In this exercise, \( \sqrt{x^2+3} \) is such an expression.

To simplify these limits, we leverage algebraic techniques like multiplying and dividing by the same term, which was \( x \) in our problem. We simplify the denominator \( \frac{\sqrt{x^2+3}}{x} \) to \( \sqrt{1+\frac{3}{x^2}} \). This takes advantage of the properties of square roots and powers.

• As \( x \) grows large, the term \( \frac{3}{x^2} \) shrinks to 0.
• This simplifies the denominator to \( \sqrt{1+0} = 1 \).

With the simplification using \( x \), the square root no longer complicates our limit evaluation, making the expression easier to solve: \( 2 + 0 = 2 \). Ultimately, this strategy helps find the limit without the square root posing any additional difficulties.