Problem 17 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

We need to evaluate the limit \( \lim _{x \rightarrow-\infty} \exp(x) \). In this problem, \( \exp(x) \) is the exponential function, which can also be written as \( e^x \), where \( e \) is the base of the natural logarithm.

Step 2: Analyze the Behavior of Exponential Function

The exponential function \( e^x \) grows fast as \( x \) increases towards positive infinity. However, we are considering \( x \) approaching negative infinity here. It's important to note that as \( x \) becomes very large and negative, \( e^x \) produces values that approach zero.

Step 3: Compute the Limit

As \( x \rightarrow -\infty \), \( e^x \rightarrow 0 \). This is because any positive number raised to a very large negative power results in a value very close to zero.

Step 4: Conclude the Result

Thus, the limit of the exponential function as \( x \) approaches negative infinity is zero. We have: \( \lim _{x \rightarrow -\infty} e^x = 0 \).

Key Concepts

Understanding the Exponential Function

The exponential function is a fundamental concept in mathematics, especially when dealing with limits, calculus, and natural logarithms. Its general form is written as \( e^x \), where \( e \) is approximately 2.71828, known as Euler's number. This number is the base of the natural logarithm, and it has unique properties making the exponential function essential in various fields of science and engineering.
When you see \( e^x \), it represents an exponential growth situation when \( x \) is positive. The function increases sharply, creating a curve that rapidly rises to the right on a graph.
Conversely, when \( x \) is negative, \( e^x \) approaches values very close to zero. This occurs because raising a positive number to a negative power results in a fraction (or very small number), which we observe as the exponential function declining steeply to the left.

• Properties: The exponential function is always positive whether \( x \) is positive or negative.
• Behavior: As \( x \) increases, the function grows, and as \( x \) decreases, it approaches zero.
• Applications: Commonly used in computing interest, population growth models, and decay processes.

Concept of Infinity in Mathematics

Infinity is a concept that represents something without any limit or end. In mathematics, it is used to describe a value that grows endlessly large, positive or negative. Its symbol is \( \infty \).
When analyzing the limit \( \lim_{x \to -\infty} e^x \), we deal with the concept of negative infinity. This indicates how the function behaves as \( x \) takes on increasingly large negative values.

• Infinite behavior: When taking limits as \( x \to \infty \) (positive infinity), functions often grow without bounds. But for \( x \to -\infty \), functions like \( e^x \) can approach zero or other finite limits.
• Imaginary aspect: Infinity isn't a number you can reach. It's rather a concept describing behavior towards an unachievable endpoint.

Thus, with limits involving infinity, we're studying how a function approaches a particular value, like zero, as the input becomes extremely large or small.

Natural Logarithm and Its Relationship to Infinity

The natural logarithm is the inverse function of the exponential function. Denoted as \( \ln(x) \), it means "the logarithm to the base \( e \)." This relationship makes it a crucial part of understanding how exponential functions behave, particularly when exploring limits and infinity.
The natural logarithm grows very slowly compared to linear or quadratic functions. Its graph rises to infinite values as \( x \) increases, though at a slow rate, reflecting its gradual growth.

• Inverse relation: If \( y = e^x \), then \( x = \ln(y) \). This shows how intertwined the natural log is with exponential growth and decay.
• Important property: Natural logarithms transform products into sums, which is useful for simplifying complex calculations: \( \ln(xy) = \ln(x) + \ln(y) \).
• Interaction with infinity: As \( x \) approaches zero from the positive side, \( \ln(x) \) approaches negative infinity, reflecting its inverse nature compared to exponential functions.

Understanding this connection is critical for grasping how exponential functions and their limits behave near infinity.