Problem 30 - 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) \(.\) $$ \lim _{u \rightarrow a}[f(u)+3 g(u)]^{3} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Laws

To solve limits, we use limit laws. The rules are that the limit of a sum is the sum of the limits, and the limit of a constant times a function is the constant times the limit of the function. Moreover, the limit of a power is the power of the limit.

Step 2: Individual Limits Evaluation

Given that \( \lim_{x \to a} f(x) = 3 \) and \( \lim_{x \to a} g(x) = -1 \), we can first compute the individual limits for \( f(u) \) and \( g(u) \) as \( u \to a \).- \( \lim_{u \to a} f(u) = 3 \)- \( \lim_{u \to a} 3g(u) = 3 \times (-1) = -3 \)

Step 3: Combine the Limits Inside the Brackets

We find \( \lim_{u \to a} [f(u) + 3g(u)] \) by combining the limits:- \( \lim_{u \to a} f(u) + \lim_{u \to a} 3g(u) = 3 + (-3) = 0 \)

Step 4: Apply Limit to the Power

Now apply the limit law for powers. Since we have found that \( \lim_{u \to a} [f(u) + 3g(u)] = 0 \),- \( \lim_{u \to a} [f(u) + 3g(u)]^3 = (\lim_{u \to a} [f(u) + 3g(u)])^3 = 0^3 = 0 \)

Key Concepts

Limit Laws

Limit laws are essential tools for evaluating limits in calculus. These set of rules guide us in breaking down complex expressions into simpler parts when limits are involved.
The primary limit laws include:

• The limit of a sum is equal to the sum of individual limits, which means if you're taking the limit of an expression like \( f(x) + g(x) \), you can separately find the limit of \( f(x) \) and \( g(x) \) and then add them up.
• The limit of a constant multiplied by a function is the constant times the limit of the function. If you have a constant \( c \) and a function \( f(x) \), the limit law states that \( \lim_{x \to a} cf(x) = c \cdot \lim_{x \to a} f(x) \).
• Finally, the limit of a power of a function is the power of the limit of the function. This is the case when you have a function raised to a power, like \( [f(x)]^n \), and it translates to \( [\lim_{x \to a} f(x)]^n \).

Understanding these basic properties can simplify complex problems and help find solutions more easily. Using these rules significantly reduces the complexity of evaluating limits.

Evaluation of Limits

Evaluating limits involves applying the limit laws to simplify the expressions until the limit can be evaluated directly. In the given problem, we know the individual limits of the functions:

• \( \lim_{x \to a} f(x) = 3 \)
• \( \lim_{x \to a} g(x) = -1 \)

With these, we can start by evaluating each component of the expression separately. For example, we can compute the limit of \( 3g(u) \) by using the constant multiple rule:

• \( \lim_{u \to a} 3g(u) = 3 \times (-1) = -3 \)

Next, we combine these results to evaluate the limit inside the brackets:

• \( \lim_{u \to a} [f(u) + 3g(u)] = 3 + (-3) = 0 \)

By breaking down through individual limits and adding them, you obtain a simplified form, hence making the process far less intimidating.

Power of Limits

The concept of power of limits is particularly useful when dealing with expressions raised to an exponents. According to the power rule in limit laws, if a function \( g(x) \) approaches a limit \( L \) as \( x \to a \), then the limit of \( [g(x)]^n \) is \( L^n \).
In this exercise, once we reached the expression \( \lim_{u \to a} [f(u) + 3g(u)] = 0 \), applying the power of limits rule becomes straightforward:

• \( \lim_{u \to a} [f(u) + 3g(u)]^3 = [\lim_{u \to a} (0)]^3 = 0^3 \)

This results in \( 0 \), as raising zero to any positive power still yields zero. Understanding and applying the power of limits helps in solving problems where expressions are compounded by exponents. Remember, ensuring that the underlying function approaches a specific limit can direct this process smoothly.