Problem 18 - Calculus for Biology and Medicine

Question

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \exp \left[-x^{2}\right] $$

Step-by-Step Solution

Answer

The limit is 0 as \( x \to \infty \).

Step 1: Identify the Function

First, let's identify the function whose limit we need to evaluate. We have the function \( \exp(-x^2) \). This is the exponential function with the argument \( -x^2 \).

Step 2: Analyze the Behavior as x Approaches Infinity

Consider the behavior of \( -x^2 \) as \( x \rightarrow \infty \). The term \( x^2 \) grows rapidly towards infinity, thus \( -x^2 \) becomes a very large negative number.

Step 3: Examine Exponential Function Behavior

The function \( \exp(u) \) where \( u \) is a large negative number approaches zero. As \( -x^2 \to -\infty \), \( \exp(-x^2) \rightarrow 0 \). This is because the exponential of any very large negative number approaches zero.

Step 4: Conclude the Limit

Since \( \exp(-x^2) \rightarrow 0 \) as \( x \rightarrow \infty \), the limit is zero. Therefore, the final result of the limit evaluation is \( 0 \).

Key Concepts

Exponential Function

The exponential function is a fundamental concept in calculus and mathematics. It's commonly represented as \( \exp(x) \) or \( e^x \), where \( e \) is Euler's number, approximately 2.71828. This function is special because it grows very rapidly if the exponent \( x \) is positive.
In our problem, we have \( \exp(-x^2) \). Here the argument of the exponential function, \( -x^2 \), is negative. This flips the scenario, greatly affecting the behavior of the function. The exponential function decreases rapidly as the exponent becomes more negative.
Understanding the exponential function's behavior is key in evaluating limits, especially when the exponent is based on more complex terms like \( -x^2 \).

Behavior at Infinity

The behavior of functions as \( x \to \infty \) is a crucial concept in calculus. It helps us understand how functions behave as they go toward extremely large values. In our exercise, we examine \( \exp(-x^2) \) as \( x \to \infty \).

• As \( x \) grows, \( x^2 \) becomes very large.
• This makes \( -x^2 \) a very large negative number.
• Since exponential functions with large negative exponents tend toward zero, \( \exp(-x^2) \to 0 \).

This behavior demonstrates how powerful the principles of infinity can be in determining limits of complex expressions. Recognizing what happens "at infinity" can simplify problem-solving in calculus.

Calculus Concepts

Calculus offers a diverse set of tools to analyze functions and their behavior, and limits are among the most fundamental. Limits describe the value a function approaches as the input approaches some point, often infinity. In our exercise, we're particularly interested in \( \lim_{x \to \infty} \exp(-x^2) \).
When solving limits, especially involving exponential functions, we often utilize these important calculus concepts:

• **Limit Laws**: These include rules like \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \), which can simplify computation.
• **Behavior of Exponentials**: The exponential function's growth or decay rates are crucial. If the exponent is large and positive, the function grows; if negative, it decays.
• **Infinity**: Functions approaching infinity often behave differently based on their nature, as demonstrated with \( -x^2 \) leading to a negative exponential.

Understanding these fundamental calculus concepts allows for the effective evaluation of limits, facilitating deeper insights into the behavior of complex functions as they approach extreme values.