Problem 1 - Calculus: Early Transcendentals

Question

Explain in your own words the meaning of each of the following. (a) \( \displaystyle \lim_{x \to \infty} f(x) = 5 \) (b) \( \displaystyle \lim_{x \to - \infty} f(x) = 3 \)

Step-by-Step Solution

Answer

(a) As \( x \to \infty \), \( f(x) \) approaches 5. (b) As \( x \to -\infty \), \( f(x) \) approaches 3.

Step 1: Understanding Limits at Infinity

When an expression such as \( \lim_{x \to \infty} f(x) = 5 \) is given, it describes the behavior of the function \( f(x) \) as \( x \) becomes very large, positively approaching infinity. Similarly, \( \lim_{x \to -\infty} f(x) = 3 \) signifies that \( f(x) \) approaches 3 as \( x \) goes to negative infinity.

Step 2: Analyzing Part (a)

The expression \( \lim_{x \to \infty} f(x) = 5 \) indicates that as \( x \) increases without bound (approaching infinity), the values of \( f(x) \) get closer and closer to 5. This implies that we should expect the function's output or value to stabilize around 5 for very large values of \( x \).

Step 3: Analyzing Part (b)

The expression \( \lim_{x \to -\infty} f(x) = 3 \) means that as \( x \) decreases without bound (approaching negative infinity), the values of \( f(x) \) get closer and closer to 3. This suggests that the function's output is expected to stabilize around 3 for very large negative values of \( x \).

Key Concepts

Behavior of Functions

Understanding the behavior of functions is essential in analyzing how they respond or change as their input values grow large or small. It gives us insight into the

• consistency
• stability
• trends of the function's output

For instance, if a function's limit is defined as the input approaches infinity, we can predict the output's stabilization point.

This stabilization occurs because the function's output values become closer to a particular number.

In the case of our example functions, as their inputs grow enormous, they approach a specific constant. This concept helps in predicting and understanding long-term trends within mathematical models.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as the input value moves towards infinity or negative infinity. This behavior is pivotal in establishing

• how close two lines are going to be
• whether the function will continue approaching a certain value indefinitely

It provides insight into the "end behavior" of a function and typically involves horizontal asymptotes when these limits lead to a specific constant value.

For instance, when a function is said to have a limit at infinity, we might visualize it approaching an imaginary horizontal line (asymptote) that it will never actually touch. This helps in comprehending the function's ultimate value target without infinite computation.

Infinite Limits

The term "infinite limits" describes how a function behaves as the input variable approaches infinity—either positively or negatively. It emphasizes whether the function reaches a

• finite value
• grows without bound
• descends infinitely

In our example problems, the specific limits like \( \lim_{x \to \infty} f(x) = 5 \), represent finite limits achieved even as the input becomes unboundedly large.

This remarkable relationship is valuable for understanding steady states in mathematical environments or verifying the feasibility of mathematical models. Moreover, it indicates the maximum or minimum long-term projected value a function can achieve.