Problem 55 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

\( \lim_{x \to -1} G(x) = 1 \); \( \lim_{x \to -2} G(x) \) does not exist.

Step 1: Understand the Function

The function given is \( G(x) = \frac{1}{x+2} \). This is a rational function with a potential vertical asymptote where the denominator is zero, i.e., at \( x = -2 \).

Step 2: Graph the Function

Graphing \( G(x) = \frac{1}{x+2} \) shows that there is a vertical asymptote at \( x = -2 \) and the curve is hyperbolic, approaching zero as \( x \) moves away from \(-2\).

Step 3: Calculate \( \lim_{x \to -1} G(x) \)

Substitute \( x = -1 \) into \( G(x) \): \( G(-1) = \frac{1}{-1+2} = 1 \). Thus, \( \lim_{x \to -1} G(x) = 1 \) because we are using a valid input and there are no discontinuities at this point.

Step 4: Calculate \( \lim_{x \to -2} G(x) \)

Approaching from the right, as \( x \to -2^+ \), \( G(x) \to +\infty \); from the left, as \( x \to -2^- \), \( G(x) \to -\infty \). Since the left and right limits are not equal, \( \lim_{x \to -2} G(x) \) does not exist.

Key Concepts

Rational Functions

Rational functions are a type of function represented by the ratio of two polynomials. In the expression \( G(x) = \frac{1}{x+2} \), the function is composed of a numerator and a denominator, where the numerator is a simple constant value (1), and the denominator is the linear polynomial \(x+2\). The behavior of these functions can be complex due to the potential for division by zero, leading to undefined points, or discontinuities, in their graphs.

• For a rational function to be defined at a certain point, the denominator must not be zero.
• These functions often exhibit vertical asymptotes where the denominator equals zero.
• They can also possess horizontal or oblique asymptotes, depending on the degrees of the polynomials involved.

Understanding the behavior of a rational function helps in predicting the graph's shape and identifying undefined points such as vertical asymptotes.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never crosses. These occur in rational functions when the denominator equals zero, creating a point of discontinuity. For \( G(x) = \frac{1}{x+2} \), the vertical asymptote is identified at \( x = -2 \).

• At \( x = -2 \), the function is undefined as the denominator reaches zero, creating a vertical asymptote.
• The function behaves differently on either side of this asymptote, often rising to infinity on one side and falling to negative infinity on the other.
• This behavior is indicative of a discontinuity, meaning the function does not have a limit at the point of the asymptote.

Recognizing vertical asymptotes is crucial for accurately graphing rational functions and understanding their limits.

Graphing Functions

Graphing functions like \( G(x) = \frac{1}{x+2} \) provides a visual representation of their behavior across the coordinate plane. The graph of a rational function can reveal asymptotes, intercepts, and the overall shape of the curve. The key steps in graphing such a function include:

• Identify vertical asymptotes and other discontinuities.
• Determine the function's behavior near these asymptotes, observing how the curve approaches them.
• Plot several points to understand how the function behaves between and beyond asymptotes.
• Look for horizontal asymptotes, if applicable, by considering the limits of \( G(x) \) as \( x \) approaches infinity or negative infinity.

The graph of \( G(x) = \frac{1}{x+2} \) showcases a classic hyperbolic curve that nears the asymptote without touching it, expands outwards, and stretches infinitely in its specified directions. Graphing not only aids in visual learning but also enhances comprehension of functional limits and asymptotic behavior.