Problem 60 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ g(x)=\frac{1}{x+2}+4 ; \quad \text { find } \lim _{x \rightarrow \infty} g(x) \text { and } \lim _{x \rightarrow-2} g(x). $$

Step-by-Step Solution

Answer

\( \lim_{x \to \infty} g(x) = 4 \); \( \lim_{x \to -2} g(x) \) does not exist.

Step 1: Analyze the Function

The given function is \( g(x) = \frac{1}{x+2} + 4 \). This function consists of a rational part \( \frac{1}{x+2} \) with a vertical asymptote at \( x = -2 \) and a horizontal translation upward by 4 units.

Step 2: Determine the Limit as x Approaches Infinity

As \( x \to \infty \), the fraction \( \frac{1}{x+2} \to 0 \). Thus, the function \( g(x) = \frac{1}{x+2} + 4 \to 0 + 4 = 4 \). Therefore, \( \lim_{x \to \infty} g(x) = 4 \).

Step 3: Determine the Limit as x Approaches -2

As \( x \to -2 \), the denominator of the fraction \( \frac{1}{x+2} \) approaches 0. Approaching from the right (\( x \to -2^+ \)), the fraction \( \frac{1}{x+2} \to +\infty \), meaning \( g(x) \to +\infty \). Approaching from the left (\( x \to -2^- \)), the fraction \( \frac{1}{x+2} \to -\infty \), meaning \( g(x) \to -\infty \). Consequently, \( \lim_{x \to -2} g(x) \) does not exist.

Key Concepts

Rational Functions

Rational functions are expressions of the form \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. These functions are intriguing because the divisor, \( q(x) \), can introduce points of discontinuity. These discontinuities typically occur when \( q(x) = 0 \), causing the function to be undefined at those values of \( x \). This leads to vertical asymptotes or holes, based on the behavior of \( p(x) \). For example, in our exercise, the function was expressed as \( g(x) = \frac{1}{x+2} + 4 \). Here, the rational part \( \frac{1}{x+2} \) defines important characteristics of the function. With \( x+2 \) in the denominator, setting it to zero \( (x+2=0) \) indicates a vertical asymptote at \( x = -2 \). Understanding the structure of rational functions helps analyze their behavior as \( x \) approaches certain critical points.

Asymptotes

Asymptotes are lines that a function approaches but never quite reaches. They act like invisible boundaries in the graph of a function. Asymptotes can be:

• Vertical
• Horizontal
• Oblique

Vertical asymptotes occur where the denominator goes to zero, making the expression in the function undefined at that point. In the exercise's function \( g(x) = \frac{1}{x+2} + 4 \), we identified a vertical asymptote at \( x = -2 \) due to \( x+2 \) in the denominator. Another type to consider is a horizontal asymptote, which describes the behavior of the function as \( x \to \infty \) or \( x \to -\infty \). In our rational function, the horizontal asymptote is at \( y = 4 \), reflecting the constant term when the fraction component \( \frac{1}{x+2} \to 0 \) as \( x \to \infty \). Observing these asymptotes provides a clearer understanding of a function's long-term behavior, making it easier to sketch and interpret its graph.

Limit Analysis

Limit analysis involves understanding the behavior of a function as it approaches a particular point or infinity. This concept is crucial in calculus as it helps define function continuity, differentiability, and infinite behavior. To find \( \lim_{x \to \infty} g(x) \) for \( g(x) = \frac{1}{x+2} + 4 \), we observe that as \( x \to \infty \), \( \frac{1}{x+2} \to 0 \). Naturally, \( g(x) \to 4 \). The horizontal asymptote at \( y = 4 \) confirms this limit behavior. When evaluating \( \lim_{x \to -2} g(x) \), we encounter the vertical asymptote at \( x = -2 \). As \( x \to -2^+ \), \( \frac{1}{x+2} \to +\infty \) which means \( g(x) \to +\infty \). Conversely, as \( x \to -2^- \), \( \frac{1}{x+2} \to -\infty \), hence \( g(x) \to -\infty \). The different directions from either side imply that the limit does not exist at this point. Mastering these techniques in limit analysis aids in comprehending the nuances of function behavior and its graphical interpretation.