Problem 22 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to \infty}\frac{x^2}{\sqrt{x^4 + 1}} \)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Rewrite the Expression

We start by rewriting the limit expression as \( \lim_{x \to \infty} \frac{x^2}{\sqrt{x^4 + 1}} \). This fraction is a rational function, which we can simplify by factoring within the square root.

Step 2: Factor the Square Root

Notice that we can factor out \( x^4 \) from the square root in the denominator: \( \sqrt{x^4 + 1} = \sqrt{x^4(1 + \frac{1}{x^4})} = x^2\sqrt{1 + \frac{1}{x^4}} \). This simplifies our limit to: \( \lim_{x \to \infty} \frac{x^2}{x^2 \sqrt{1 + \frac{1}{x^4}}} \).

Step 3: Simplify the Expression

Cancel out \( x^2 \) from the numerator and the denominator: \( \lim_{x \to \infty} \frac{x^2}{x^2 \sqrt{1 + \frac{1}{x^4}}} = \lim_{x \to \infty} \frac{1}{\sqrt{1 + \frac{1}{x^4}}} \).

Step 4: Evaluate the Limit

As \( x \to \infty \), the term \( \frac{1}{x^4} \to 0 \). Therefore, the expression inside the square root becomes \( \sqrt{1 + 0} = 1 \). So, the limit is \( \lim_{x \to \infty} \frac{1}{1} = 1 \).

Key Concepts

Rational Functions

Rational functions are expressions that involve the ratio of two polynomials. They are of the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. In our exercise, the rational function is \( \frac{x^2}{\sqrt{x^4 + 1}} \). Here, the polynomial in the numerator is \( x^2 \) and the denominator contains a square root expression with polynomials involved.
Understanding rational functions is essential because they often involve limits and asymptotic behavior. They can be simplified using algebraic techniques like factorization, which can make finding limits more straightforward. Notice how the denominator \( \sqrt{x^4 + 1} \) is factored to reveal simpler expressions for higher value calculations. This factorization is critical when evaluating limits of rational functions as they approach infinity.

Asymptotic Behavior

Asymptotic behavior refers to the behavior of a function as the input either increases or decreases without bound. It's a way to describe how functions behave at their extremes.
In our problem, we're particularly interested in the limit as \( x \to \infty \). The rational function \( \frac{x^2}{\sqrt{x^4 + 1}} \) simplifies as \( x \) becomes very large. By factorizing, it becomes \( \frac{1}{\sqrt{1 + \frac{1}{x^4}}} \).

• At \( x \rightarrow \infty \), \( \frac{1}{x^4} \rightarrow 0 \).
• The expression becomes \( \sqrt{1 + 0} = 1 \).

So, the original rational function approaches 1 as \( x \rightarrow \infty \). This behavior is a vital concept in calculus to predict the limits at asymptotes or infinity.

Factorization

Factorization is a technique used to simplify expressions or solve equations by expressing a number or an expression as a product of its factors.
In our original problem, we employed factorization to simplify \( \sqrt{x^4 + 1} \). This was rewritten as \( \sqrt{x^4(1 + \frac{1}{x^4})} = x^2\sqrt{1 + \frac{1}{x^4}} \).

• The factor \( x^4 \) is taken out to simplify the expression under the square root.
• This allows us to cancel matching terms in both numerator and denominator.

Factorization is critical in dealing with polynomials and is a gateway to simplifying complex expressions which otherwise might seem challenging to evaluate directly.

Square Roots

Square roots are often encountered in calculus, especially dealing with polynomials. A square root \( \sqrt{x} \) refers to the value that, multiplied by itself, produces \( x \).
In the problem, the denominator involves \( \sqrt{x^4 + 1} \). Understanding how to work with square roots is essential:

• Square roots distribute over multiplication, which helps in factorization.
• It's crucial to simplify expressions like \( \sqrt{x^4 + 1} \) to \( x^2 \sqrt{1 + \frac{1}{x^4}} \) to more easily cancel terms and find limits.

Thus, breaking down square roots in algebraic limits helps in defining the behavior of functions at extremes.