Problem 14 - Calculus Early Transcendentals

Question

\(13-14\) Evaluate the limit and justify each step by indicating the appropriate properties of limits. $$ \lim _{x \rightarrow \infty} \sqrt{\frac{12 x^{3}-5 x+2}{1+4 x^{2}+3 x^{3}}} $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Identify the Highest Power Terms

Look at the numerator and the denominator of the expression inside the square root, \( \frac{12x^3 - 5x + 2}{1 + 4x^2 + 3x^3} \). Identify the highest power of \( x \) in both the numerator and the denominator. The highest power in both is \( x^3 \).

Step 2: Factor Out the Highest Power of x

Factor \( x^3 \) from both the numerator and the denominator: \( \frac{x^3(12 - \frac{5}{x^2} + \frac{2}{x^3})}{x^3(\frac{1}{x^3} + \frac{4}{x} + 3)} \). This is justified because the limit behavior is determined by the highest degree terms as \( x \to \infty \).

Step 3: Simplify the Limit Expression

Cancel \( x^3 \) from the numerator and denominator, which simplifies to \( \frac{12 - \frac{5}{x^2} + \frac{2}{x^3}}{\frac{1}{x^3} + \frac{4}{x} + 3} \).

Step 4: Evaluate the Limit as x Approaches Infinity

Take the limit as \( x \to \infty \). Terms with \( x \) in the denominator approach zero, so \( \frac{5}{x^2} \to 0 \), \( \frac{2}{x^3} \to 0 \), \( \frac{1}{x^3} \to 0 \), and \( \frac{4}{x} \to 0 \). Thus the expression becomes \( \frac{12}{3} \).

Step 5: Calculate the Square Root

Take the square root of the result in step 4: \( \sqrt{\frac{12}{3}} = \sqrt{4} = 2 \). This is the final value of the limit.

Key Concepts

Asymptotic Behavior

The concept of asymptotic behavior helps us to describe how a function behaves as the input approaches a particular value, often at infinity. It’s like predicting how a function behaves when you zoom out very far. In our exercise, as the value of \( x \) becomes extremely large, the lower degree terms like \(-5x\) and \(+2\) vanish in significance compared to the cubic term \(12x^3\). The same goes for the denominator with terms like \(+4x^2\) and \(+1\), which become negligible against \(3x^3\). Asymptotic behavior allows us to simplify complex functions into much simpler forms, focusing only on the highest power terms. This simplification is crucial for evaluating limits as \( x \rightarrow \infty \). This is why the exercise instructs us to factor out the highest powers, focusing on the terms that dominate the growth of the function at large \( x \). This technique gives a clearer picture of how numerator and denominator behave in relation to each other asymptotically.

Limit Properties

Limit properties are essential rules and techniques that help in simplifying and solving limit problems. They allow us to break down complex expressions and analyze limit behavior systematically. Some essential properties include:

• Quotient Property: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.
• Constant Factor Rule: Constants can be brought out of the limit.
• Sum/Difference Rule: The limit of a sum or difference is the sum or difference of the limits.

In our step-by-step solution, limit properties are applied after simplifying the expression. For instance, we used the quotient property to divide the limits of numerator and denominator. The constant factor rule allows us to factor and simplify the function by focusing on the highest power term in \( x \). Once all terms with \( x \) in the denominator tend toward zero, basic limit properties enable us to further simplify the expression to find the final value.

Infinity Limits

Infinity limits explore the behavior of a function as it approaches positive or negative infinity. They help us in understanding how a function progresses as it drifts off towards extremes, providing insights about a function’s growth or decay. In evaluating the limit as \( x \rightarrow \infty \), we focus on the most significant terms. By canceling the highest degree term in both the numerator and the denominator, we're simplifying the exploration to analyze how each portion behaves when \( x \) becomes exceedingly large.Infinity limits involve ignoring smaller, less significant terms, as these become irrelevant when compared to much larger dominating terms. The function simplifies drastically, allowing us to strip away all terms that diminish to zero, finally leading to a clean evaluation of the function's behavior at extreme values.