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: Identify Given Limits

We are given that \( \lim_{x \to a} f(x) = 3 \). This means as \( x \) approaches \( a \), the function \( f(x) \) approaches the value 3.

Step 2: Recognize Expression Structure

The limit we need to find is \( \lim_{x \to a}[f(x) - 3]^{4} \). Alternately, this can be expressed as raising the term \( [f(x)-3] \) to the fourth power.

Step 3: Substitute the Limit into the Expression

Since \( \lim_{x \to a} f(x) = 3 \), the expression \( f(x) - 3 \) approaches 0 as \( x \to a \). Therefore, \( \lim_{x \to a} (f(x) - 3) = 0 \).

Step 4: Apply Limits to Power Rule

Raised to the fourth power, \( [f(x) - 3]^{4} \) becomes 0 raised to the fourth power as \( x \) approaches \( a \). This follows the power rule: \( \lim_{x \to a} (u(x))^{n} = (\lim_{x \to a} u(x))^{n} \) if the limit \( u(x) \) exists.

Step 5: Calculate the Final Limit

Applying the power rule to our expression, we have \( \lim_{x \to a} [f(x)-3]^{4} = (\lim_{x \to a} [f(x)-3])^{4} = 0^{4} = 0 \).

Key Concepts

Power rule in calculus

The power rule in calculus provides a straightforward way to differentiate functions that are exponentials of a term. However, it is also key in the context of evaluating limits of expressions that involve powers. The rule states that, if you have a function raised to an exponent, you can evaluate its limit by firstly finding the limit of the base and then raising that result to the given power. Formally, this is written as:\[ \lim_{x \to a} (u(x))^{n} = \left( \lim_{x \to a} u(x) \right)^{n} \]where the limit \( u(x) \) exists.

• To apply the power rule, ensure the base function’s limit is well-defined and finite. Otherwise, the power rule may not apply.
• This rule is useful in simplifying expressions and computing limits that would otherwise seem complex without it.
• In our exercise, the rule allowed us to simplify \( \lim_{x \to a} [f(x)-3]^{4} \) to \( 0^4 = 0 \), since the limit of \( f(x)-3 \) was 0.

Whenever you encounter a limit involving a powered expression, think of the power rule as your go-to strategy to simplify your calculations.

Continuity in calculus

Continuity is a fundamental property of functions in calculus that ensures a function behaves predictably around certain points. A function is continuous at a point if the following conditions are simultaneously met:

• The function is defined at that specific point.
• The limit of the function as it approaches the point is defined.
• The value of the function at that point is equal to the limit of the function as it approaches the point.

In simpler terms, a continuous function will have no breaks, jumps, or holes at the point of interest.For our given exercise, understanding continuity is key because continuity allows us to assert that as \( x \) approaches \( a \), the behavior of \( f(x) \) smoothly transitions through the point that \( f(x) \) would approach its limit. Because \( \lim_{x \to a} f(x) = 3 \), if \( f(x) \) is continuous at \( x = a \), this transition is seamless.

Evaluating limits

Evaluating limits involves finding the value a function approaches as the input approaches a specific point. This process can seem daunting, but there are several strategies and rules, including the power rule, that simplify this task immensely.

• Direct Substitution: Apply direct substitution if the function is continuous at the point of interest. It often provides the limit directly.
• Simplifying Expressions: Factor or use algebraic identities to simplify the expression before evaluating the limit.
• L'Hôpital's Rule: Use when you encounter indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In the exercise, the limit \( \lim_{x \to a} [f(x)-3]^{4} \) required recognizing it as a power of a simple expression. By substituting the known limit \( \lim_{x \to a} (f(x) - 3) = 0 \), we were able to swiftly compute the final limit. Understanding these strategies equips you with the toolkit to tackle a variety of limit problems with confidence.