Problem 1 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{x \rightarrow \infty} 2 /(x-3) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Type of Limit

The limit we are asked to evaluate is \( \lim_{x \to \infty} \frac{2}{x - 3} \). This is a rational function, specifically one that approaches a horizontal asymptote as \( x \) tends to infinity.

Step 2: Analyze the Denominator's Behavior

As \( x \) approaches infinity, the expression \( x - 3 \) within the denominator becomes very large (positively infinite in value).

Step 3: Consider the Overall Expression's Behavior

For any constant number like \( 2 \), when it is divided by a number that becomes infinitely large, the result approaches zero. Hence, \( \frac{2}{x-3} \rightarrow 0 \) as \( x \rightarrow \infty \).

Step 4: Compute the Limit

Based on the steps above, the expression \( \frac{2}{x - 3} \) tends towards zero as \( x \) approaches infinity. Therefore, the limit is \( 0 \).

Key Concepts

Rational Functions

Rational functions are expressions that can be written as the fraction of two polynomials. They are of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions, and \( Q(x) eq 0 \). In our exercise, the rational function is \( \frac{2}{x - 3} \). Here, the numerator is a constant polynomial \( 2 \), and the denominator is a linear polynomial \( x - 3 \).
Understanding rational functions is important for solving limits as they can behave differently based on their numerator and denominator degrees. For instance:

• If the degree of the numerator is less than the degree of the denominator, as \( x \to \infty \), the limit approaches zero.
• If the degree of the numerator and denominator are equal, the limit is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, the expression tends towards infinity.

In many rational function limits like this one, analyzing the behavior as \( x \to \infty \) or \( x \to -\infty \) can tell us a lot about the behavior of the graph and the asymptotic tendencies.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as \( x \to \pm\infty \). They help describe the end behavior of a function. In rational functions, horizontal asymptotes are determined by the degrees of the polynomials in the numerator and the denominator.
For a rational function such as \( \frac{2}{x-3} \):

• The numerator (constant 2) has degree 0.
• The denominator \( x-3 \) has degree 1.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \( y = 0 \).
This tells us that as \( x \to \infty \) or \( x \to -\infty \), the values of the rational function will get closer and closer to 0 but never actually touch or cross the line \( y = 0 \). This illustrates a critical concept in understanding how rational functions behave at their extremes.

Infinite Limits

Infinite limits occur when the value of a function becomes arbitrarily large or small as \( x \) approaches a particular value. When dealing with limits at infinity like \( \lim _{x \rightarrow \, \infty} f(x) \), we are interested in the behavior of the function as \( x \) grows larger and larger.
In the given exercise, the limit in question is \( \lim _{x \rightarrow \, \infty} \frac{2}{x-3} \):

• As \( x \rightarrow \, \infty \), the denominator \( x-3 \) becomes very large, while the numerator \( 2 \) remains constant.
• Dividing a constant by an increasingly large number yields a value that becomes closer and closer to zero.

Thus, in this specific calculation, the limit resolves to 0. Recognizing how values tend to grow or shrink as \( x \to \infty \) in rational functions can often provide insights into their long-term behavior and, subsequently, their horizontal asymptotes.