Problem 103 - Essential Calculus

Question

In Exercises \(101-104,\) use the definition of limits at infinity to prove the limit. $$ \lim _{x \rightarrow-\infty} \frac{1}{x^{3}}=0 $$

Step-by-Step Solution

Answer

Using the definition of limits at infinity, we have proved that \(\lim_{x \to -\infty} \frac{1}{x^3} = 0\).

Step 1: Understand the Definition of Limits at Infinity

The formal definition of limit at infinity states that, for a function \(f(x)\), \(\lim_{x \to -\infty} f(x) = L\) means that for every real number \(\epsilon > 0\), there is a real number \(N < 0\) such that if \(x < N\), then \(|f(x) - L| < \epsilon\).

Step 2: Apply the Definition to the Given Function

Applying the definition to the function, \(\lim_{x \to -\infty} \frac{1}{x^3} = 0\) means that for every real number \(\epsilon > 0\), there exists a real number \(N < 0\) such that if \(x < N\), then \(|\frac{1}{x^3} - 0| = |\frac{1}{x^3}| < \epsilon\). It can also be simplified to \(|-x^3| > \frac{1}{\epsilon}\), which results in \(x < - \sqrt[3]{\frac{1}{\epsilon}}\). Here, the real number \(N\) is \(- \sqrt[3]{\frac{1}{\epsilon}}\).

Step 3: Prove the Limit is 0

For any given \(\epsilon > 0\), if \(x < - \sqrt[3]{\frac{1}{\epsilon}}\) (which is \(N\)), it guarantees that \(|\frac{1}{x^3} - 0| < \epsilon\). Therefore, we can conclude that \(\lim_{x \to -\infty} \frac{1}{x^3} = 0\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. Within this field, limits are a fundamental concept, particularly when analyzing how functions behave as inputs approach a particular value or infinity.

In our context, calculus enables us to assess the behavior of a function as we look far out towards negative or positive infinity. For functions that extend towards infinity, calculus offers tools such as limits to describe their asymptotic behavior. As we consider the limit of \( \frac{1}{x^3} \) as \( x \) approaches negative infinity, we are essentially exploring how this function behaves on a large negative scale through the lens of calculus.

Definition of Limit

The definition of a limit is a way to describe the value that a function approaches as the input approaches a certain point. For finite points, this means finding the function's output value as \( x \) approaches a specific number. A limit can exist even if the function's value isn't defined at that point.

However, when discussing limits at infinity, we are interested in the value that a function approaches as \( x \) gets larger and larger in magnitude (either positively or negatively). In the case of the exercise provided, we're exploring what happens to \( \frac{1}{x^3} \) as \( x \) decreases without bound (or goes to negative infinity), which involves using the formal definition of limits at infinity.

Infinite Limits

Infinite limits refer to the situation where a function's value grows without bound as the input either gets very large or very small. It may approach a positive or negative infinity or it may approach a finite limit, such as the case where a function approaches zero.

Understanding infinite limits also involves knowing how to prove them rigorously using \( \epsilon \) and \( N \) as demonstrated in the solution. The proof is structured to show that no matter how small an \( \epsilon \) we choose (representing a band around the limit value), we can always find a corresponding threshold \( N \) in the domain of \( x \) that guarantees the function’s output will be within this band. The exercise follows these steps to prove that as \( x \) goes to negative infinity, \( \frac{1}{x^3} \) approaches zero, illustrating the concept of an infinite limit where the value of the function decreases down to zero as \( x \) becomes infinitely negative.