Question

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 0}\left(e^{2 x}-1\right)=0 $$

Short answer

Set \( \delta = \frac{\varepsilon}{2} \), then \( |e^{2x} - 1| < \varepsilon \) for \( |x| < \delta \).

Step-by-step solution

Understand the Limit Statement

We need to prove that \( \lim_{x \to 0} (e^{2x} - 1) = 0 \) using an \( \varepsilon-\delta \) proof. This means that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x| < \delta \), it follows that \( |e^{2x} - 1| < \varepsilon \).

Identify the \( \varepsilon-\delta \) Condition

For the definition of the limit, we need to ensure \( |e^{2x} - 1| < \varepsilon \) for \( |x| < \delta \). Our task is to express \( \delta \) in terms of \( \varepsilon \).

Simplify the Expression

Let's analyze \( |e^{2x} - 1| \). Using the property of the exponential function, we can approximate \( e^{2x} \) using its series expansion: \( e^{2x} = 1 + 2x + \frac{(2x)^2}{2!} + \cdots \). For small \( x \), \( e^{2x} \approx 1 + 2x \). So, \( |e^{2x} - 1| \approx \left| 2x \right| = 2|x| \).

Relate \( |x| \) to \( \varepsilon \)

From the approximation \( |e^{2x} - 1| \approx 2|x| \), we want \( 2|x| < \varepsilon \). Therefore, we choose \( |x| < \frac{\varepsilon}{2} \).

Define \( \delta \)

We know from the previous step that \( |x| < \frac{\varepsilon}{2} \). Therefore, we can set \( \delta = \frac{\varepsilon}{2} \). This satisfies the \( \varepsilon-\delta \) definition of a limit.

Conclude the Proof

For every \( \varepsilon > 0 \), we can choose \( \delta = \frac{\varepsilon}{2} \). With this \( \delta \), whenever \( 0 < |x| < \delta \), it follows that \( |e^{2x} - 1| < \varepsilon \). This completes the \( \varepsilon-\delta \) proof that \( \lim_{x \to 0} (e^{2x} - 1) = 0 \).

Key concepts

Limit of a Function

The concept of the limit of a function is foundational in understanding calculus. When we talk about finding the limit of a function as it approaches a certain point, we're essentially trying to determine what value the function is getting closer to as the input, or variable, approaches a particular number. This is a crucial skill in calculus as it helps in understanding function behavior, especially when dealing with derivatives and integrals.

The formal definition involves using \(\varepsilon-\delta\) proofs, where for any desired level of closeness, or \(\varepsilon > 0\), there exists a small interval around the point (defined by \(\delta\)) such that the function's value remains within this target range as the input nears the specific point. In other words:

• If we choose any small positive number \(\varepsilon\), there exists another small positive number \(\delta\) such that for every x within the interval \(0 < |x - a| < \delta\), the distance between \(f(x)\) and the limit is less than \(\varepsilon\).
• This ensures that as x approaches a specific value, the function's output gets infinitely close to a particular limit.

For the exercise, the goal was to prove the limit \(\lim_{x \to 0} (e^{2x} - 1) = 0\) using this \(\varepsilon-\delta\) concept. Understanding this principle allows us to explore how subtle changes in input affect the function and helps in grounding more complex calculus concepts.

Exponential Functions

Exponential functions are a key part of algebra and calculus. These functions involve terms that have a variable as the exponent, typically with the base \(e\), which is approximately 2.71828. This unique mathematical constant, known as Euler's number, is used extensively in calculus due to its natural properties.

An exponential function can be represented as \(e^{ax}\), where \(a\) determines the rate of growth or decay, and \(x\) is the variable. Exponential functions exhibit rapid growth or decay based on the sign and size of \(a\):

• If \(a > 0\), the function exhibits exponential growth.
• If \(a < 0\), it exhibits exponential decay.

In calculus, exponential functions are important because they describe many natural phenomena, like population growth and radioactive decay.

In the exercise given, we're focused on \(e^{2x}\). For small values of \(x\), the exponential function can be approximated using a series expansion, allowing us to simplify the problem for easier calculation:

• For instance, \(e^{2x} \approx 1 + 2x\) when \(x\) is close to zero.
• This approximation helps in managing calculus proofs related to exponential functions.

Understanding the properties of exponential functions is essential when performing tasks like derivatives, integration, or \(\varepsilon-\delta\) proofs as they often appear in complex calculus equations.

Calculus Proofs

Calculus proofs are methods used to rigorously demonstrate the truth of mathematical statements within calculus. One of the common proofs is the \(\varepsilon-\delta\) proof, which is essential in establishing the limits of functions. Proofs in calculus are critical as they validate theorems and provide a structured way to derive or demonstrate key mathematical concepts.

To conduct an \(\varepsilon-\delta\) proof, several steps must be followed:

• Start by understanding what you need to prove and clearly define your target limit and function.
• Translate your problem into an \(\varepsilon-\delta\) statement, ensuring that for any small \(\varepsilon > 0\) there exists a \(\delta > 0\) such that the function’s behavior falls within the limits as described.
• Simplify the function where possible, often using approximations, like the Taylor series expansion, to make calculations manageable.
• Relate the simplified expression to \(\varepsilon\), find an appropriate \(\delta\), and validate the initial \(\varepsilon-\delta\) condition.

For the given exercise, an \(\varepsilon-\delta\) proof was used to show that \(\lim_{x \rightarrow 0}(e^{2x}-1)=0\). This involved simplifying \(e^{2x}\) to approximately \(1 + 2x\) for small \(x\), allowing us to express \(\delta\) in terms of \(\varepsilon\) as half of \(\varepsilon\).This meticulous step-by-step approach assures us that the proof is solid and logical. Understanding how these proofs work is crucial for mastering calculus.