Problem 10 - Precalculus: Mathematics for Calculus

Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 0}\left(3 x^{3}-2 x^{2}+5\right)$$

Step-by-Step Solution

Answer

The limit is 5.

Step 1: Apply Limit Law for Sums

The limit of a sum is the sum of the limits. So, apply the sum rule for the limit \( \lim_{x \to 0}(3x^3 - 2x^2 + 5) \) to get \( \lim_{x \to 0}(3x^3) - \lim_{x \to 0}(2x^2) + \lim_{x \to 0}(5) \).

Step 2: Apply Limit Law for Constants and Powers

Using the constant multiple rule (\( \lim_{x \to 0}(c \times f(x)) = c \times \lim_{x \to 0} f(x) \)) and power rule (\( \lim_{x \to 0}(x^n) = 0 \) for \( n > 0 \)), evaluate each term: \( 3 \times \lim_{x \to 0}(x^3) - 2 \times \lim_{x \to 0}(x^2) \). Given that the limits of \( x^3 \) and \( x^2 \) as \( x \to 0 \) are both 0, these calculations yield \( 3 \times 0 - 2 \times 0 \).

Step 3: Evaluate Constant Limit

According to the limit laws, \( \lim_{x \to 0}(5) \) is simply 5, because the limit of a constant is the constant itself.

Step 4: Combine the Results

Combine all the results from the previous steps: \( 3 \times 0 - 2 \times 0 + 5 = 5 \).

Key Concepts

Constant Multiple Rule

When evaluating limits, one crucial concept is the Constant Multiple Rule. This rule assists when you encounter a term in a function that is multiplied by a constant value. For example, consider the function term \(3x^3\) in the expression \(3x^3 - 2x^2 + 5\). The Constant Multiple Rule can be expressed as follows:

• \( \lim_{x\to a} (c \cdot f(x)) = c \cdot \lim_{x\to a} f(x) \)

This means we can multiply the constant by the limit of the function without affecting the limit solution itself.
The Constant Multiple Rule simplifies the process, allowing us to work with the limit of the function directly, alleviating the complexity introduced by the constant. As shown in the step-by-step solution, applying this rule enables simplified calculation: \(3 \cdot \lim_{x \to 0} (x^3)\).
This speeds up calculations by addressing constants before dealing with functions.

Sum Rule for Limits

The Sum Rule for Limits is another helpful approach when handling the limit of a function composed of several parts added or subtracted together. This rule states that the limit of a sum or difference of functions is the sum or difference of their limits.

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

Applying this rule helps in breaking down complex expressions into more manageable parts.
As in the original step-by-step solution: \( \lim_{x \to 0} (3x^3 - 2x^2 + 5)\) is split into \( \lim_{x \to 0} (3x^3) - \lim_{x \to 0} (2x^2) + \lim_{x \to 0} (5)\).
This makes it simpler to calculate each part separately, especially when different components involve constants and powers, aiding in step-by-step evaluation.

Power Rule for Limits

The Power Rule for Limits is particularly beneficial when dealing with polynomial expressions, such as terms like \(x^n\). This rule helps evaluate limits as the variable approaches a specified value, often zero.

• \( \lim_{x\to a} x^n = a^n \)

For cases where the limit involves \(x\) approaching zero, the rule becomes even simpler: \( \lim_{x \to 0} x^n = 0\) for \(n > 0\).
This was applied in the solution when evaluating parts like \( \lim_{x \to 0} x^3 = 0\) and \( \lim_{x \to 0} x^2 = 0\).
The Power Rule streamlines the process, making it straightforward to find limits of polynomial terms, especially when they'd equal zero as the variable vanishes. In combination with the Constant Multiple Rule and Sum Rule, it forms a robust toolkit for evaluating limits.