Problem 8 - Calculus: Early Transcendentals

Question

For the limit $$ \lim_{x \to 0}\frac{e^{2x} - 1}{x} = 2 $$ illustrate Definition 2 by finding values of \( \delta \) that correspond to \( \varepsilon = 0.5 \) and \( \varepsilon = 0.1 \).

Step-by-Step Solution

Answer

For \( \varepsilon = 0.5 \), \( \delta = 0.25 \); for \( \varepsilon = 0.1 \), \( \delta = 0.05 \).

Step 1: Understand the Definition of a Limit

The formal definition of a limit states that for \( \lim_{x \to a} f(x) = L \), for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \varepsilon \). In this problem, we need to find \( \delta \) for which \( \left| \frac{e^{2x} - 1}{x} - 2 \right| < \varepsilon \).

Step 2: Substitute and Simplify

Substitute the given functions and the limit value into the definition: \( \left| \frac{e^{2x} - 1}{x} - 2 \right| < \varepsilon \). Simplify the inequality: \[ \left| \frac{e^{2x} - 1 - 2x}{x} \right| < \varepsilon. \] This can further be simplified to \( |e^{2x} - 1 - 2x| < \varepsilon |x| \).

Step 3: Estimate \( \delta \) for \( \varepsilon = 0.5 \)

For \( \varepsilon = 0.5 \), we want \( |e^{2x} - 1 - 2x| < 0.5|x| \). For small \( x \), use the approximation \( e^{2x} \approx 1 + 2x + 2x^2 \), yielding \( |2x^2| < 0.5|x| \). Solving \( |2x| < 0.5 \) gives \( |x| < 0.25 \). Thus, choose \( \delta = 0.25 \).

Step 4: Estimate \( \delta \) for \( \varepsilon = 0.1 \)

Now, solve the inequality \( |e^{2x} - 1 - 2x| < 0.1|x| \) with the approximation earlier, \( |2x^2| < 0.1|x| \), leading to \( |2x| < 0.1 \). Solving gives \( |x| < 0.05 \). So, choose \( \delta = 0.05 \).

Step 5: Conclusion

According to the steps above, for \( \varepsilon = 0.5 \), we can choose \( \delta = 0.25 \) and for \( \varepsilon = 0.1 \), \( \delta = 0.05 \). These values of \( \delta \) show that the limit satisfies the definition given specific \( \varepsilon \) values.

Key Concepts

Limit of a Function

The concept of the limit of a function is foundational in calculus. Limits allow us to understand the behavior of a function as the input (or variable) approaches a certain point. Essentially, the limit tells us what value the function is getting closer to even if it never quite reaches that value. In the problem above, for \( \lim_{x \to 0}\frac{e^{2x} - 1}{x} = 2 \), the limit is examined as \( x \) approaches zero.

Here's how it works:

• We are interested in what happens to \( \frac{e^{2x} - 1}{x} \) as \( x \to 0 \).
• The result should be that this expression gets closer and closer to 2.
• Mathematically, we confirm this using the epsilon-delta definition of a limit, seeking a \( \delta \) for each \( \varepsilon \), to satisfy \( \left| \frac{e^{2x} - 1}{x} - 2 \right| < \varepsilon \).

This set-up helps us confirm or define precisely when and how a function approaches a particular value, assuring us the function follows a predictable path as it nears a designated point.

Exponential Function

Exponential functions, such as \( e^{2x} \), play a crucial role in calculus due to their unique properties. These functions are defined by having a constant base raised to a variable exponent, and they exhibit fast growth or decay.

Key Characteristics:

• Rapid Growth or Decay: As the exponent grows larger, the value of the function increases rapidly, depending on whether the base is greater or less than one.
• Continuous and Differentiable: Exponential functions are smooth and continuous, making them easy to analyze and integrate into calculus problems.
• E = Euler's Number: The base \( e \), approximately 2.718, is a prevalent choice, being the natural exponential constant due to properties like \( \frac{d}{dx} e^x = e^x \).

In our problem, we used the exponential function to demonstrate the limit process. By approximating \( e^{2x} \) as \( 1 + 2x + 2x^2 \) for small values of \( x \), it becomes easier to simplify and manage calculations.

Approximations in Calculus

Approximations are indispensable tools in calculus, helping us simplify complex expressions and make calculations manageable. When dealing with limits, especially, we often approximate functions to see how components behave as they become infinitely small or large.

Considerations Include:

• Simplifying Expressions: We use approximations like \( e^{2x} \approx 1 + 2x + 2x^2 \) to reduce the complexity of exponential expressions.
• Highlighting Main Term Contributions: When \( x \) is small, higher-order terms (like \( 2x^2 \) in our problem) contribute less to the overall function, thus can often be omitted for easier computations.
• Building Intuition: These approximations don't just ease calculations but help us intuitively understand how functions behave — seeing first-hand how changes in \( x \) affect the overall function gives powerful insights.

In mathematical studies, approximations are valuable for confirming limits, estimating derivatives or integrals, and reducing intricate formulas into simpler forms to better grasp underlying patterns and behaviors.