Problem 5 - Calculus Early Transcendentals

Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 8}(1+\sqrt[3]{x})\left(2-6 x^{2}+x^{3}\right)$$

Step-by-Step Solution

Answer

The limit is 390.

Step 1: Apply the Limit to Each Factor

First, we need to apply the limit separately to each factor inside the expression. The given expression is \(\left(1+\sqrt[3]{x}\right)\left(2-6x^{2}+x^{3}\right)\). By the Fundamental Limit Law, \(\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\), provided that both limits exist. So, start with \(\lim_{x \to 8} \left(1+\sqrt[3]{x}\right)\) and \(\lim_{x \to 8} \left(2-6x^{2}+x^{3}\right)\).

Step 2: Evaluate the First Limit

Evaluate \(\lim_{x \to 8} \left(1+\sqrt[3]{x}\right)\). By the Sum and Root Laws of limits, this can be split into two separate limits: \(\lim_{x \to 8} 1 + \lim_{x \to 8} \sqrt[3]{x}\). The limit of a constant is the constant itself, so \(\lim_{x \to 8} 1 = 1\). The limit \(\lim_{x \to 8} \sqrt[3]{x} = \sqrt[3]{8} = 2\). Together, they sum up to \(1 + 2 = 3\).

Step 3: Evaluate the Second Limit

Evaluate \(\lim_{x \to 8} \left(2 - 6x^{2} + x^{3}\right)\). By applying the Sum and Power Laws of limits, first break it down: \(\lim_{x \to 8} 2 - 6 \lim_{x \to 8} x^{2} + \lim_{x \to 8} x^{3}\). \(\lim_{x \to 8} 2 = 2\), \(\lim_{x \to 8} x^{2} = 8^{2} = 64\), and \(\lim_{x \to 8} x^{3} = 8^{3} = 512\). Calculate the expression: \(2 - 6 \cdot 64 + 512 = 2 - 384 + 512 = 130\).

Step 4: Multiply the Results

Combine the results of the two limits according to the first step. Multiply the limits from Step 2 and Step 3 together: \(3 \times 130 = 390\). This product is the value of the original limit as \(x\) approaches 8.

Key Concepts

Limit Laws

Limit laws are a set of foundational rules in calculus that outline how limits operate. They make evaluating complex limits feel almost straightforward. When dealing with limits, we can apply these laws to simplify expressions, often breaking them into more manageable parts. Key limit laws include:

• Constant Law: The limit of a constant is the constant itself. In symbols, \(\lim_{{x \to a}} c = c\).
• Sum Law: Allows breaking down limits into separate, simpler parts: \(\lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)\).
• Product Law: States that the limit of a product is the product of the limits, given both exist: \(\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)\).
• Power Law: Useful when limits involve powers of functions: \(\lim_{{x \to a}} [f(x)]^n = [\lim_{{x \to a}} f(x)]^n\).

These laws collectively simplify the evaluation by reducing complicated expressions into known values or more familiar forms.

Power Law

In calculus, the Power Law for limits is essential when dealing with expressions that contain powers of a variable or function. This law simplifies the process by allowing you to "move" the exponent outside of the limit
For example, if you need to evaluate the limit of \(x^3\) as \(x\) approaches a particular value, the limit simply becomes that value raised to the power of three:
\(\lim_{{x \to a}} x^n = a^n\).
In the given exercise, we find the limit of \(x^2\) and \(x^3\) as \(x\) approaches 8. Applying the Power Law gives us:

• \(\lim_{{x \to 8}} x^2 = 8^2 = 64\)
• \(\lim_{{x \to 8}} x^3 = 8^3 = 512\)

Both results are straightforward using the Power Law, showing how powerful and effective it is for quickly solving these limits.

Sum Law

The Sum Law allows us to evaluate limits of sums by separately considering each part of the sum. It can simplify calculations dramatically. This is particularly useful when the function can be broken down into a sum of individual terms.
For example, if you have a limit like \(\lim_{x \to a}[f(x) + g(x)]\), the Sum Law lets you evaluate it simply as:

• \(\lim_{x \to a} f(x) + \lim_{x \to a} g(x)\)

In the exercise, this law is used to evaluate limits individually for the expression \(2 - 6x^2 + x^3\).

• Start with \(\lim_{x \to 8} 2\), giving a constant limit of 2.
• Then,\(\lim_{x \to 8} -6x^2\), which uses the Power Law for each part: \(6 \times 64\)
• Finally, \(\lim_{x \to 8} x^3\), calculated using the power law, becomes 512.

These separate limits neatly combine into \(2 - 384 + 512\), resulting in a final value of 130, all simplified using the Sum Law together with other limit laws.

Root Law

The Root Law of limits connects closely to the Power Law, specifically when dealing with roots or radical expressions. It states that taking a limit of a root is equivalent to taking the root of the limit.
If you encounter an expression like \(\sqrt[n]{x}\), and you need to find its limit as \(x\) approaches a value, you can compute it as follows:
\(\lim_{{x \to a}} \sqrt[n]{x} = \sqrt[n]{a}\).
In the exercise, the Root Law helps in evaluating the term \(\lim_{x \to 8} \sqrt[3]{x}\). It becomes:

• \(\sqrt[3]{8} = 2\)

The Root Law simplifies this process, turning what might seem like a daunting task into a simple calculation. By using this law, we establish that the challenging radical expression is quite manageable, providing us with an essential tool for limit evaluation.