Problem 37 - Calculus for Biology and Medicine

Question

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} e^{-x^{2} / 3} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Limit Definition

The problem asks for the limit of the function \( e^{-x^{2}/3} \) as \( x \) approaches 0. This means we need to find the value the function approaches when \( x \) gets very close to 0.

Step 2: Substitute the Limit Point into the Function

Substitute \( x = 0 \) directly into the function: \( e^{-0^{2}/3} = e^{0} \).

Step 3: Evaluate the Exponential Expression

Since \( e^{0} = 1 \), the value of the function \( e^{-x^{2}/3} \) at \( x = 0 \) is 1.

Step 4: Conclude with the Limit Result

Thus, the limit is \( \lim _{x \rightarrow 0} e^{-x^{2}/3} = 1 \).

Key Concepts

Exponential Functions

Exponential functions are a vital component of mathematics, especially when dealing with calculus and limits. The function that we are looking at, \( e^{-x^2/3} \), is a typical example of such an exponential expression.
In this function, \( e \) stands for Euler's number, approximately equal to 2.71828. It's the base of natural logarithms and is crucial for many growth and decay processes.

Why do exponential functions play such a significant role in calculus? Mainly because of their unique growth properties and their continuous, smooth nature.

• They can represent real-world phenomena like population growth and radioactive decay.
• They remain smooth and continuous, making them ideal for analyzing change and rates.

In our problem, the function involves \( e \) raised to a power that includes a quadratic term (\(-x^2/3\)). This indicates a rapidly changing value as \( x \) varies. Understanding the properties of \( e \) helps in evaluating the limit smoothly, showcasing the blend of algebra and calculus with exponential functions.

Limit Evaluation

Evaluating limits is a foundational concept in calculus. A limit describes the behavior of a function as its input approaches a particular value. In our exercise, we're finding \( \lim_{x \to 0} e^{-x^2/3} \).

Limits help us understand the behavior of functions at points where direct evaluation is difficult or impossible. In contexts where a function doesn't behave as expected (like division by zero), limits offer insight into what happens as we get closer to an interesting point.
In this specific problem:

• We're interested in the behavior of the function \( e^{-x^2/3} \) as \( x \) gets closer and closer to 0.
• The challenge is to determine what value the expression tends towards.

By plugging \( x = 0 \) into the function, we eliminate the variable dependency and directly find what the function reaches, effectively solving the limit. Recognizing when substitution is possible is key to swiftly evaluating limits like these.

Substitution Method

The substitution method is often the go-to technique for evaluating limits. As seen in the solution steps, substituting \( x = 0 \) into the function \( e^{-x^2/3} \) provides a straightforward path to finding the limit.

This approach works well when substituting directly doesn't lead to undefined forms such as \( \frac{0}{0} \). Here, you can:

• Simplify the expression by replacing the variable with the given limit point.
• If after substitution the expression is still defined, it gives the LIM value directly.

For our function, substituting \( x = 0 \) results in \( e^{0} \), which simplifies to 1. This determination is straightforward because the exponential function is continuous everywhere, including at \( x = 0 \).

Remember, the substitution method is effective in many real-world situations where you need to quickly gauge the behavior of a function at a particular point, especially when the function is continuous.