Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { . Prove that } \lim _{x \rightarrow c}|x|=|c| \text { . } $$

Step-by-Step Solution

Answer

The limit is \(|c|\), because \(| |x| - |c| |\) is less than any \(\varepsilon\) for \(|x - c| < \varepsilon\).

Step 1: Understand the Problem Statement

We need to prove that the limit of the absolute value function \(|x|\) as \(x\) approaches \(c\) is equal to the absolute value of \(c\), that is, \( \lim_{x \to c} |x| = |c| \).

Step 2: Recall the Definition of Absolute Value

The absolute value of a number \(x\), denoted \(|x|\), is \(x\) if \(x \geq 0\) and \(-x\) if \(x < 0\). This function is continuous everywhere.

Step 3: Definition of a Limit

According to the definition of a limit, \( \lim_{x \to c} f(x) = L \) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \varepsilon\).

Step 4: Apply the Limit Definition to |x|

For \(|x|\), we want \(| |x| - |c| | < \varepsilon \) whenever \(0 < |x - c| < \delta \). Start by observing that \(| |x| - |c| | \leq |x - c| \).

Step 5: Choose \(\delta\) and Conclude

By our observation, \(| |x| - |c| | \leq |x - c|\). Thus, setting \(\delta = \varepsilon\) ensures that \(| |x| - |c| | < \varepsilon \) whenever \(0 < |x - c| < \delta\), which proves the limit is \(|c|\).

Key Concepts

Definition of Limit

In mathematics, understanding the concept of a limit is crucial when studying how functions behave as they approach a specific point. Specifically, when we say that the limit of a function \( f(x) \) as \( x \) approaches some point \( c \) is \( L \), denoted as \( \lim_{x \to c} f(x) = L \), we mean that \( f(x) \) gets arbitrarily close to \( L \) as \( x \) gets closer and closer to \( c \).

Here's what's happening in simpler terms:

• We look at what happens to \( f(x) \) very near \( x = c \) and make sure \( f(x) \) can get as close as we like to \( L \).
• For every tiny positive distance, let's call it \( \varepsilon \), between \( f(x) \) and \( L \), there should be a range (\( \delta \)) where \( x \) stays close to \( c \) and ensures \( f(x) \) within that distance.

To put it plainly, the function hugs the value \( L \) more tightly as \( x \) zeroes in on \( c \). It's like a game of horseshoes, getting closer every step to the target.

Absolute Value Function

The absolute value function is a simple yet essential concept in mathematics. Denoted by \(|x|\), it measures how far a number is from zero on the number line. Here's how it works:

• If \( x \) is a positive number or zero, \(|x|\) is simply \( x \).
• If \( x \) is negative, then \(|x|\) is the positive opposite of \( x \), or \(-x\).

This function effectively strips a number of its sign, leaving you with its magnitude or size.

For example:

• |3| = 3 (because 3 is positive)
• |-5| = 5 (because -5 is negative, so the positive opposite is 5)

The absolute value function is continuous, meaning its graph has no breaks, jumps, or holes. It's like drawing a smooth hill on paper. A continuous function is valuable as its output doesn't suddenly change for small movements in \( x \). This feature provides a sturdy foundation for proving limit properties.

Continuity of Functions

Continuous functions are vital in calculus and analysis because they don't have sudden jumps or interruptions. Importantly, they precisely maintain their values when variables slightly shift, making limits more straightforward to manage.

For a function to be continuous at a point \( c \), it satisfies these criteria:

• The function \( f(x) \) is defined at \( c \), which means \( f(c) \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( c \) exists, denoted as \( \lim_{x \to c} f(x) \).
• The limit \( \lim_{x \to c} f(x) \) equals the function value \( f(c) \).

With these fulfilled, the graph of the function \( f(x) \) will be a smooth curve at that point, without abrupt changes.Considering the absolute value function \(|x|\), it's continuous everywhere on the real number line. This means no matter what point you pick, the function doesn't surprise you with a jump. So, when we prove a limit involving \(|x|\), like \( \lim_{x \to c} |x| = |c| \), continuity makes this process seamless. It's as if we drew the entire \(|x|\) function graph in one motion.