Problem 29 - 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) \(.\) $$ \lim _{t \rightarrow a}[|f(t)|+|3 g(t)|] $$

Step-by-Step Solution

Answer

The limit is 6.

Step 1: Identify the given limits

We are given that \( \lim _{x \rightarrow a} f(x) = 3 \) and \( \lim _{x \rightarrow a} g(x) = -1 \). These represent the limits of two functions \( f(x) \) and \( g(x) \) as \( x \) approaches \( a \).

Step 2: Apply limit properties

When finding the limit of a sum or difference, you can evaluate the limits individually. Therefore, \( \lim _{t \rightarrow a}[|f(t)|+|3 g(t)|] = \lim _{t \rightarrow a}|f(t)| + \lim _{t \rightarrow a}|3 g(t)| \).

Step 3: Evaluate \(\lim _{t \rightarrow a}|f(t)|\)

Since \( \lim _{x \rightarrow a} f(x) = 3 \), the limit of \( |f(t)| \) as \( t \rightarrow a \) is the absolute value of 3. Thus, \( \lim _{t \rightarrow a}|f(t)| = |3| = 3 \).

Step 4: Evaluate \(\lim _{t \rightarrow a}|3 g(t)|\)

The expression \( 3g(t) \) can be rewritten as \( 3 \times g(t) \), and \( \lim _{x \rightarrow a} g(x) = -1 \), so \( \lim _{t \rightarrow a} 3g(t) = 3 \times (-1) = -3 \). The absolute value of \( -3 \) is 3, hence \( \lim _{t \rightarrow a}|3 g(t)| = 3 \).

Step 5: Combine the limits

Add the limits obtained in the previous steps: \( \lim _{t \rightarrow a} [|f(t)| + |3 g(t)|] = 3 + 3 = 6 \).

Key Concepts

Limit Properties

Calculus limits can sometimes feel daunting, but understanding basic limit properties makes working with them much easier. Let's dive into some of these properties that can simplify the process of finding limits.

**Key Limit Properties:**

• **Sum/Difference Rule:** When you have the limit of a sum or a difference, you can break it down like so: if you know that \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} [f(x) \pm g(x)] = L \pm M \).
• **Constant Multiple Rule:** If you're working with a constant multiplier, the limit of a constant times a function is just the constant times the limit of the function. For example, \( \lim_{x \to a} [k \cdot f(x)] = k \cdot L \).
• **Absolute Value Rule:** The limit of an absolute value is the absolute value of the limit. This is often used when dealing with absolute values in limits.

By using these properties, you can decompose complex limits into smaller, more manageable parts. This makes it much easier to evaluate each part separately and then combine them back together for your final answer.

Absolute Values in Limits

Understanding absolute values in limits is crucial for mastering calculus limits. Absolute values can change the nature of a problem since they take the non-negative value of what's inside. Here's how to deal with them.

**Calculating Limits with Absolute Values:**

• When considering the limit of an absolute value, simply take the absolute value of the limit itself. For instance, if \( \lim_{x \to a} f(x) = L \), then \( \lim_{x \to a} |f(x)| = |L| \).
• The absolute value removes any negative sign from the final value. By effectively 'measuring' the magnitude, you negate any directional component the original value had.
• In the example provided, \( \lim_{t \to a} |f(t)| \) and \( \lim_{t \to a} |3g(t)| \) were calculated as the absolute values of 3 and -3, respectively, both turning into 3.

Recognizing when and how to apply absolute values can make a big difference in solving such limits. It ensures that you correctly account for any changes that could affect the outcome of the original function as it approaches the limit.

Function Limits

Function limits are the foundational idea behind calculus limits, helping us understand the behavior of a function as it approaches a specific point. Grasping this concept unlocks deeper insights into calculus.

**Approaching a Point:**

• When evaluating \( \lim_{x \to a} f(x) \), you're seeking to discover how the function \( f(x) \) behaves as \( x \) draws near to \( a \). It's about understanding the function's behavior at that precise moment.
• The concept doesn’t necessarily require the function to actually reach \( f(a) \). Instead, it evaluates the pattern of the function’s proximity as \( x \) increasingly approaches \( a \).
• This understanding allows us to predict and define the behavior of more complex or "undefined" areas of a function, giving us a mathematical way to observe continuity and discontinuity.

Mastering function limits empowers you with the ability to navigate and analyze the continuous changes and trends a function can showcase as it nears certain values or points. This is essential in exploring wider calculus topics efficiently.