Problem 27 - Calculus Early Transcendentals

Question

Give proofs of the following theorems. a. \(\lim _{x \rightarrow a} c=c \quad\) for any constant \(c\) b. \(\lim _{x \rightarrow a} x=a\) for any constant \(a\)

Step-by-Step Solution

Answer

Question: Prove the following limit theorems: (a) The limit of a constant function as x approaches any value is the constant itself, and (b) the limit of the identity function as x approaches any constant value is the same constant value. Answer: (a) We have proved that for any constant function c, the limit as x approaches any value a is the constant itself: \(\lim_{x \rightarrow a} c = c\). (b) We have also proved that for the identity function x, the limit as x approaches any constant value a is the same constant value: \(\lim_{x \rightarrow a} x = a\).

Step 1: Proof of Theorem (a): Limit of a constant function

Given the limit theorem: \(\lim _{x \rightarrow a} c=c\) for any constant \(c\). We need to show that, given any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all x satisfying \(0 < |x - a| < \delta\), \(|c - c| < \epsilon\). Since \(c\) is a constant, we know that \(|c - c| = 0\). Thus, no matter the value of \(x\), \(|c - c|\) is always \(0\). Now, \(0\) is always less than or equal to any positive value of \(\epsilon\), so we can choose any \(\delta > 0\) to make this inequality true: \(0 < |x - a| < \delta \implies |c - c| < \epsilon\) Therefore, \(\lim _{x \rightarrow a} c=c\) for any constant \(c\).

Step 2: Proof of Theorem (b): Limit of the identity function

Given the limit theorem: \(\lim _{x \rightarrow a} x=a\) for any constant \(a\). We need to show that, given any \(\epsilon > 0\), there exists a \(\delta\) > 0 such that for all x satisfying \(0 < |x - a| < \delta\), the inequality \(|x - a| < \epsilon\) holds. Since we want to find a \(\delta\) that satisfies the above inequality, we can directly choose \(\delta = \epsilon\). Therefore, for all x satisfying \(0 < |x - a| < \delta\), the inequality \(|x - a| < \epsilon\) holds. Thus, we have proved that \(\lim _{x \rightarrow a} x=a\) for any constant \(a\).

Key Concepts

Constant Function

A constant function is one of the simplest types of functions out there. The key characteristic of a constant function is that its output remains the same for any input value. This is expressed as a function of the form \( f(x) = c \), where \( c \) is some constant value. Imagine drawing a horizontal line on a graph; no matter where you move along the x-axis, the value of \( f(x) \) stays the same—it's just \( c \).

In terms of limits, the limit of a constant function as \( x \) approaches \( a \) is simply the constant value itself. Here's why:

• The function doesn't change as \( x \) changes, which means \(|c-c|=0\), making our epsilon (\(\epsilon\)) choice quite flexible.
• For any positive \(\epsilon\), \(|c-c|=0\) is automatically less than any positive \(\epsilon\).

Thus, we can choose any positive delta (\(\delta\)) because the difference \(|c-c|\) will always meet our condition for the limit definition. This simplicity is why constant functions provide an intuitive introduction to the concept of limits.

Identity Function

An identity function is where inputs equal outputs. It's represented by \( f(x) = x \). For each input \( x \), the output is the same. On a graph, if you plot \( x \) against \( f(x) \), the graph forms a diagonal line passing exactly through the origin at a 45-degree angle.When considering limits for the identity function as \( x \) tends toward some constant \( a \), the function behaves as follows:

• We want \(|x-a|\) to be less than some small number \(\epsilon\).
• To achieve this, we can directly set \(\delta = \epsilon\) since \(|x-a|\) directly simplifies our limit condition.

This reflects the characteristic straightforwardness of the identity function, offering a basic yet effective demonstration of how limits work.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to define what it means for a function to approach a limit. It might sound a bit technical at first, but it's an essential tool in calculus. Let's break it down for better understanding:Imagine \( f(x) \) approaches a limit \( L \) as \( x \) approaches \( a \). In epsilon-delta terms:

• For every small number \(\epsilon\) (epsilon), there is a small number \(\delta\) (delta) such that whenever \(|x-a| < \delta\), it guarantees that \(|f(x) - L| < \epsilon\).
• This means we can make \( f(x) \) as close to \( L \) as we wish by choosing \( x \) values sufficiently close to \( a \).

• The choice of \(\delta\) is vital but only depends on our chosen \(\epsilon\) and how the function behaves.

Visualizing this definition helps—it’s like tightening a tolerance, honing in on the precise behavior of the function near \( a \). This method brings a precise, logical foundation to understand limits across calculus.