Problem 5 - Precalculus: Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{6}{x}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Function

The function given is \( f(x) = \frac{6}{x} \). We are tasked with evaluating \( \lim _{x \rightarrow \infty} \frac{6}{x} \).

Step 2: Understand the Behavior as \(x\) Approaches Infinity

As \(x\) becomes very large (approaches infinity), the denominator \(x\) in the expression \(\frac{6}{x}\) increases, leading the entire fraction to become smaller.

Step 3: Apply Limit Properties

Recall that \( \lim_{x \to \infty} \frac{c}{x} = 0 \) for any constant \(c\). This property arises because the numerator is constant (here, \(6\)) and the denominator grows without bound.

Step 4: Calculate the Limit

Using the property of limits, we substitute the behavior into the expression: \( \lim_{x \to \infty} \frac{6}{x} = 0 \).

Key Concepts

Infinity

When we talk about infinity, especially in relation to limits, we're considering what happens as values get extremely large. Infinity (\(\infty\)) isn't a number that can be reached, but a concept that describes unbounded growth. In calculus, when we evaluate limits as a variable approaches infinity, we're interested in understanding the behavior of a function as its input grows without bound.
For example, consider the fraction \(\frac{1}{x}\). As \(x\) becomes very large, the value of \(\frac{1}{x}\) becomes very small, approaching zero. This is because a larger denominator means that the overall value of the fraction will shrink. Understanding this behavior is crucial when dealing with limits approaching infinity, as it influences how we evaluate the expression.

Limit Properties

When working with limits, certain properties simplify the process of finding limits in various scenarios. One such property is \(\lim_{x \to \infty} \frac{c}{x} = 0\), where \(c\) is a constant. This property is especially useful when dealing with rational expressions, where the numerator is a constant and the denominator includes an \(x\) that becomes very large.This happens because as \(x\) increases, \(\frac{c}{x}\) gets smaller and smaller, effectively approaching zero. Knowing limit properties allows us to quickly determine the behavior of a function without lengthy computations. Applying these properties efficiently can help simplify complex problems into more manageable parts. This technique was used when evaluating \(\lim_{x \to \infty} \frac{6}{x}\) because the numerator is constant, making it straightforward to conclude that the limit is zero.

Rational Functions

Rational functions are expressions of the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. In calculus, analyzing the limits of rational functions as \(x\) approaches infinity is a common task. It's important to understand the roles of the numerator and the denominator.
When the degree of the polynomial in the denominator is greater than in the numerator, such as in \(\frac{6}{x}\), as \(x\) grows, the function approaches zero. This occurs because the denominator becomes much larger than the numerator, leading to smaller fractions and eventually zero as the limit.In contrast, if the numerator's degree were higher or equal to the denominator's, the behavior could be different, often resulting in constants or infinity, depending on how the degrees compare. Therefore, severely understanding rational functions and their degrees is crucial for tackling limit problems effectively.