Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1\) (see Example 4\()\). $$\lim _{x \rightarrow a}[f(x)-3]^{4}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Limit

We are given that \( \lim_{x \rightarrow a} f(x) = 3 \). We want to find \( \lim_{x \rightarrow a} [f(x) - 3]^4 \). With the limit of \( f(x) \) known, we can understand how the expression \( f(x) - 3 \) behaves as \( x \to a \).

Step 2: Applying the Known Limit

Since \( \lim_{x \rightarrow a} f(x) = 3 \), it follows that \( \lim_{x \rightarrow a} [f(x) - 3] = \lim_{x \rightarrow a} f(x) - 3 = 3 - 3 = 0 \).

Step 3: Finding the Limit of the Expression

Given that \( \lim_{x \rightarrow a} [f(x) - 3] = 0 \), the next step is to find \( \lim_{x \rightarrow a} [f(x) - 3]^4 \). By the power rule for limits, \( \lim_{x \rightarrow a} [f(x) - 3]^4 = \left( \lim_{x \rightarrow a} [f(x) - 3] \right)^4 = 0^4 = 0 \).

Key Concepts

Understanding Limit Properties

Limits are a fundamental part of calculus and help us understand how a function behaves as its input approaches a particular value. In this exercise, we are given that \( \lim _{x \rightarrow a} f(x)=3 \) and need to understand how this affects other expressions, such as \( \lim _{x \rightarrow a} [f(x)-3]^4 \).
Limit properties are helpful rules that allow us to manipulate limits in a predictable manner. These include:

• **Sum and Difference Property:** The limit of a sum or difference is the sum or difference of the limits, i.e., \( \lim_{x \to c} (f(x) \pm g(x)) = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x) \).
• **Constant Multiple Property:** For a constant \( k \), \( \lim_{x \to c} [k f(x)] = k \cdot \lim_{x \to c} f(x) \).
• **Power Property:** If you have a limit raised to a power, such as \( [f(x)]^n \), the limit of the expression is \( (\lim_{x \to c} f(x))^n \).

Applying these properties simplifies finding complex limits, like when another function or adjustment is involved in the original function.

Applying the Power Rule for Limits

The power rule for limits is an important tool when dealing with expressions that involve exponents. In this exercise, we want to find \( \lim_{x \to a} [f(x) - 3]^4 \). Knowing that \( \lim_{x \to a} f(x) = 3 \), we first compute \( \lim_{x \to a} [f(x) - 3] \), which simplifies to zero.
The power rule tells us that for a limit problem of the form \( \lim_{x \to c} [f(x)]^n \), we can compute this as \( (\lim_{x \to c} f(x))^n \). In our case:

• First, evaluate \( \lim_{x \to a} [f(x) - 3] = 0 \).
• Then apply the power rule by raising the result to the fourth power, \( 0^4 \), which equals zero.

The power rule simplifies these computations and provides a systematic approach for dealing with similar limit problems.

Evaluating Limits in Steps

Evaluating limits involves breaking down the expressions systematically to find their behavior as \( x \) approaches a certain value. For the expression \( \lim _{x \rightarrow a}[f(x)-3]^{4} \), we first use the known limit \( \lim _{x \rightarrow a} f(x) = 3 \) to simplify \( \lim_{x \rightarrow a} [f(x) - 3] \) to zero.
This leads to:

• Understanding that \( f(x) \) approaches 3, simplifying \( f(x) - 3 \) as zero when \( x \to a \).
• Then applying the power rule, taking the zero and determining \( 0^4 \), resulting in 0.

By breaking down each part of the expression, step-by-step, it becomes easier to handle even seemingly complicated limits.

Mastering Calculus Step-by-Step

Learning calculus is much like solving a puzzle: the picture becomes clearer piece by piece. This exercise shows that tackling limit problems involves understanding each component and applying the appropriate rules, such as limit properties and the power rule.
Here's a quick recap:

• Apply the limit properties to simplify expressions.
• Use known limits, like \( \lim_{x \to a} f(x) \), to break expressions into manageable parts.
• Use rules like the power rule to simplify the evaluation of limits with exponents.

Mastering calculus is achievable by recognizing patterns, practicing regularly, and breaking down each problem step by step. This approach transforms complexity into clarity.