Problem 51 - Thomas Calculus

Question

Suppose \(\lim _{x \rightarrow 0} f(x)=1\) and \(\lim _{x \rightarrow 0} g(x)=-5 .\) Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation. \begin{equation} \begin{aligned} \lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{(f(x)+7)^{2 / 3}} &=\frac{\lim _{x \rightarrow 0}(2 f(x)-g(x))}{\lim _{x \rightarrow 0}(f(x)+7)^{2 / 3}} \quad\quad\quad\quad\quad\quad\text{(a)}\\\ &=\frac{\lim _{x \rightarrow 0} 2 f(x)-\lim _{x \rightarrow 0} g(x)}{\left(\lim _{x \rightarrow 0} f(x)+7\right)^{2 / 3}}\quad\quad\quad\quad\text{(b)} \\\ &=\frac{2 \lim _{x \rightarrow 0} f(x)-\lim _{x \rightarrow 0}7^{2 / 3}}{\left(\lim _{x \rightarrow 0}f(x)+\lim _{x \rightarrow 0} 7\right)^{2 / 3}}\quad\quad\quad\quad\text{(c)} \\ &=\frac{(2)(1)-(-5)}{(1+7)^{2 / 3}}=\frac{7}{4} \end{aligned} \end{equation}

Step-by-Step Solution

Answer

The answer is \( \frac{7}{4} \).

Step 1: Apply Limit of Quotient Rule

We start with the expression \( \lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{(f(x)+7)^{2 / 3}} \). According to Theorem 1, we can apply the limit of quotient rule. This states that the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. Thus, this is rewritten as: \[ \frac{\lim _{x \rightarrow 0}(2 f(x)-g(x))}{\lim _{x \rightarrow 0}(f(x)+7)^{2 / 3}}. \]

Step 2: Apply Limit of Sum and Power Rule

For the numerator, \( \lim_{x \rightarrow 0} (2f(x) - g(x)) \), we use the limit of subtraction rule to express it as \( \lim_{x \rightarrow 0} 2f(x) - \lim_{x \rightarrow 0} g(x) \). In the denominator, \( \lim_{x \rightarrow 0} (f(x) + 7)^{2/3} \) is found by applying the power of a sum rule. This allows us to say \( \left(\lim_{x \rightarrow 0} (f(x) + 7)\right)^{2/3} \), giving us: \[ \frac{\lim_{x \rightarrow 0} 2f(x) - \lim_{x \rightarrow 0} g(x)}{\left(\lim_{x \rightarrow 0} f(x) + 7\right)^{2/3}}. \]

Step 3: Apply Limit of Constant and Identity Rule

We apply the constant factor and identity rules. For the term \( \lim_{x \rightarrow 0} 2f(x) \), apply the constant factor rule: \( 2 \cdot \lim_{x \rightarrow 0} f(x) \). For \( \lim_{x \rightarrow 0} g(x) \) and \( \lim_{x \rightarrow 0} f(x) \), apply the identity rule which gives \( -5 \) and \( 1 \) respectively. Finally, for the constant 7, \( \lim_{x \rightarrow 0} 7 \) remains 7. This yields: \[ \frac{2 \cdot 1 - (-5)}{(1 + 7)^{2/3}}. \]

Step 4: Simplify the Final Expression

Substitute the known limits into the expression. Numerator: \( 2 \times 1 - (-5) = 2 + 5 = 7 \). Denominator: \( (1 + 7)^{2/3} = 8^{2/3} = 4 \). So the limit simplifies to: \( \frac{7}{4} \).

Key Concepts

Limit of a Quotient

When dealing with limits of quotients, the limit law tells us that we can take the limit of each part of the expression separately as long as the denominator's limit does not equal zero. This is an important condition because division by zero is undefined. In our exercise, we applied this rule first by expressing the original limit as the quotient of two separate limits:

• Numerator: The limit of the expression in the numerator.
• Denominator: The limit of the expression in the denominator, provided this is not zero.

For context, if you had a quotient such as \(\frac{a(x)}{b(x)}\), and as \(x\) approaches a certain value both \(\lim_{x \rightarrow c} a(x)\) and \(\lim_{x \rightarrow c} b(x)\) exist, then:\[\lim_{x \rightarrow c} \frac{a(x)}{b(x)} = \frac{\lim_{x \rightarrow c} a(x)}{\lim_{x \rightarrow c} b(x)}\]In the exercise, this applied to the fraction \(\frac{2f(x) - g(x)}{(f(x) + 7)^{2/3}}\), allowing us to consider these limits separately.

Limit of a Sum

The limit of a sum law helps us break down expressions where multiple functions are added or subtracted. This law is very similar to adding or subtracting real numbers. If you know the limits of individual functions, you can sum or subtract them directly:

• \(\lim_{x \rightarrow c}(a(x) + b(x)) = \lim_{x \rightarrow c}a(x) + \lim_{x \rightarrow c}b(x)\)
• \(\lim_{x \rightarrow c}(a(x) - b(x)) = \lim_{x \rightarrow c}a(x) - \lim_{x \rightarrow c}b(x)\)

In our textbook solution, you saw this principle used when we had \(\lim_{x \rightarrow 0} (2f(x) - g(x))\). By employing the limit of a sum or difference, we could express this as \(\lim_{x \rightarrow 0} 2f(x) - \lim_{x \rightarrow 0} g(x)\). This separation was crucial for later steps where we substituted known values of the limits.

Limit of a Power

The limit of a power rule is applicable when we have something as a power or root within the limit expression. It essentially allows us to "pull out" and apply the power after computing the limits. The general form is:\[\lim_{x \rightarrow c} [a(x)]^n = \left(\lim_{x \rightarrow c} a(x)\right)^n\]This rule simplifies complex expressions where some component is raised to a power or involves roots, like in our example with \((f(x) + 7)^{2/3}\). By using the limit power rule, we first determine \(\lim_{x \rightarrow 0} (f(x) + 7)\), which simply involves adding constant terms since \(\lim_{x \rightarrow 0} f(x) = 1\) known from the problem. Thus, it gives us \[(\lim_{x \rightarrow 0} (f(x) + 7))^{2/3} = \left((1 + 7)\right)^{2/3}\]. This step was essential in resolving the complete expression towards a simplified form.