Problem 17 - The Calculus with Analytic Geometry

Question

Give a definition for each of the following: (a) \(\lim _{x \rightarrow+\infty} f(x)=-\infty ;\) (b) \(\lim _{x \rightarrow-\infty} f(x)=+\infty ;\) (c) \(\lim _{x \rightarrow-\infty} f(x)=-\infty\).

Step-by-Step Solution

Answer

(a) f(x) decreases without bound as x approaches +∞. (b) f(x) increases without bound as x approaches -∞. (c) f(x) decreases without bound as x approaches -∞.

Step 1: Understanding Limits at Infinity

Limits at infinity describe the behavior of a function as the input variable approaches positive or negative infinity.

Step 2: Defining \(\( \lim_{x \rightarrow+\infty} f(x) = -\infty \)\)

This definition means that as the variable x increases without bound (approaches positive infinity), the function f(x) decreases without bound (approaches negative infinity). Formally, for any large negative number M, there exists a number N such that for all x > N, f(x) < M.

Step 3: Defining \(\( \lim_{x \rightarrow-\infty} f(x) = +\infty \)\)

This definition means that as the variable x decreases without bound (approaches negative infinity), the function f(x) increases without bound (approaches positive infinity). Formally, for any large positive number M, there exists a number N such that for all x < N, f(x) > M.

Step 4: Defining \(\( \lim_{x \rightarrow-\infty} f(x) = -\infty \)\)

This definition means that as the variable x decreases without bound (approaches negative infinity), the function f(x) also decreases without bound (approaches negative infinity). Formally, for any large negative number M, there exists a number N such that for all x < N, f(x) < M.

Key Concepts

asymptotic behavior

Asymptotic behavior refers to how a function behaves as its input either grows infinitely large or infinitely small. When analyzing limits at infinity, we are essentially studying the asymptotic behavior of functions. This helps us understand the trend or direction in which the function heads as the input becomes very large or very small.

For example, if a function approaches a certain value as its input approaches infinity, we can say that the function has an asymptote at that value. Understanding the asymptotic behavior is crucial because it helps in sketching graphs and predicting long-term trends in various applications.

functions

Functions are mathematical relations where each input is mapped to exactly one output. They are often represented as f(x), where 'x' is the input variable and f(x) is the output.

In the context of limits at infinity, we usually examine how the output of a function behaves as the input becomes exceedingly large in the positive or negative direction.

This helps us understand the long-term behavior of the function. For instance, in the equations given in the exercise, we're interested in how the function f(x) behaves as x either grows to positive infinity or negative infinity.

calculus definitions

Calculus definitions provide the formal groundwork for understanding limits at infinity. When we talk about limits, especially at infinity, we use precise definitions to ensure clarity.

For example, \( \lim_{x \rightarrow+\infty} f(x) = -\infty \) means that as x grows without bound, f(x) decreases without bound. Similarly, \( \lim_{x \rightarrow-\infty} f(x) = +\infty \) means as x becomes very negative, f(x) increases without bound. These definitions help us effectively describe and analyze the behavior of functions.

This formal framework makes sure that everyone understands the exact trend and direction of a function as it moves towards infinite values.

positive infinity

Positive infinity, denoted as \(+\infty \), represents an unbounded increase in value. In simpler terms, it means going larger and larger without any limit.

When we talk about the limit of a function as x approaches positive infinity, we are examining how the function behaves as x gets very large. For instance, \( \lim_{x \rightarrow+\infty} f(x) = -\infty \) tells us that, as x becomes infinitely large, the function f(x) decreases without bound.

Positive infinity is a valuable concept in calculus to help us understand and describe the long-term behavior of functions.

negative infinity

Negative infinity, denoted as \(-\infty \), represents an unbounded decrease in value. It means going smaller and smaller without any limit.

When we talk about the limit of a function as x approaches negative infinity, we are studying how the function behaves as x becomes very large in the negative direction. For example, \( \lim_{x \rightarrow-\infty} f(x) = +\infty \) means as x becomes infinitely negative, the function f(x) increases without bound.

Understanding negative infinity is crucial for grasping the full picture of a function's behavior as it stretches towards very large negative values.