Problem 98 - Calculus of a Single Variable

Question

Proof In Exercises \(95-98\) , use the definition of limits at infinity to prove the limit. $$ \lim _{x \rightarrow-\infty} \frac{1}{x-2}=0 $$

Step-by-Step Solution

Answer

By applying the properties of limits and using the definition of limits at infinity, we have proven that \( \lim _{x \rightarrow-\infty} \frac{1}{x-2} = 0 \).

Step 1: Start with the given expression

Given function is \( \frac{1}{x-2} \). The task is to prove that as \( x \) approaches negative infinity, the limit of the function is 0.

Step 2: Apply limit properties

Looking at the function, \( \frac{1}{x-2} \), as \( x \) approaches \( -\infty \), \( x-2 \) also approaches \( -\infty \) and the whole function therefore approaches 0.

Step 3: Use the definition of limit

The definition of a limit \( \lim _{x \rightarrow-\infty} f(x) = L \) means that for every \( \epsilon > 0 \), there is a \( N<0 \) such that if \( x 0 \), we can find \( N \) such that when \( x < N \), it makes \( |\frac{1}{x-2} - 0| < \epsilon \). This \( N \) can be calculated as \( \frac{-1}{\epsilon} + 2 \). Thus it is proven that as \( x \) approaches \( -\infty \), the function approximates to 0.

Key Concepts

Definition of Limits

To understand the behavior of functions as they approach a specific input, we turn to the concept of limits—a foundational stone in calculus. A limit describes the value that a function approaches as the input approaches some value. The formal definition states that the limit of a function, usually denoted as
\[ \[ \lim_{x \to c} f(x) = L \] \],
exists if, for every number \( \epsilon > 0 \) there is a corresponding number \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \) it follows that \( |f(x) - L| < \epsilon \).
Now, when dealing with limits at infinity, the \( c \) is replaced with \(-\infty\) or \(+\infty\), and \( \delta \) is replaced with a value 'N' that signifies a particular boundary beyond which the behavior of the function is described. The concept allows us to quantify the idea of 'closeness' and 'approach' in a rigorous mathematical manner, which is essential for uncovering the properties and behavior of functions as they extend towards infinity.

Calculus

Calculus, the mathematical study of continuous change, is a branch that deals extensively with functions, limits, derivatives, and integrals. With its origins rooted in the pursuit of understanding motion and change, calculus provides the tools for analyzing and modeling the real world. It is divided into two main branches: differential calculus, concerning the rate of change and slopes of curves, and integral calculus, focusing on the accumulation of quantities and the areas under and between curves.
In understanding limits at infinity, like in our exercise \( \frac{1}{x-2} \), calculus allows us to deduce the behavior of the function when \( x \) reaches extremely large or small values. Calculus helps in making sense of how rapidly or slowly functions grow or decline, which is vital in fields ranging from physics and engineering to economics and biology.

Proofs in Calculus

Proofs in calculus provide the rigor to confirm that what intuitively seems correct also holds under formal mathematical scrutiny. When we speak about proofs involving limits, particularly those at infinity, we typically utilize the definition of a limit, limit theorems, the Squeeze theorem, or the concepts of continuity and differentiability to demonstrate certain behaviors analytically. The step-by-step solution provided illustrates a proof utilizing the formal definition of limits.
In our example, the challenge was to prove that \[ \lim _{x \rightarrow-\infty} \frac{1}{x-2}=0 \]
by establishing that as \( x \) becomes indefinitely large in the negative direction, \( \frac{1}{x-2} \) approaches zero. This sort of proof is not just mathematical abstraction; it conveys exact information about the asymptotic behavior of functions which is useful in various practical contexts, like understanding the long-term behavior of a system described by a function.

Limit Properties

The properties of limits are like the rules of the game for working with limits in calculus. They provide systematic ways to manipulate and evaluate limits without conducting the limit process each time. To mention a few key properties:

• A constant times a function: \( \lim_{x \to c} k \cdot f(x) = k \cdot \lim_{x \to c} f(x) \).
• Sum or difference of functions: \( \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x) \).
• Product of functions: \( \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \).
• Quotient of functions, provided that the limit in the denominator isn't zero: \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \).

Applying these properties can significantly simplify the process of finding the limit. They enable us to break down complex functions into simpler ones whose behavior we can readily analyze. For instance, in the step-by-step solution, we use the understanding that as \( x \) approaches \( -\infty \) the function \( \frac{1}{x-2} \) tends to zero due to its denominator growing without bounds in the negative direction.