Problem 41 - Thomas Calculus

Question

Find the limits in Exercises \(37-48.\) $$\lim _{x \rightarrow-8^{+}} \frac{2 x}{x+8}$$

Step-by-Step Solution

Answer

The limit is \(-\infty\) as \( x \rightarrow -8^{+} \).

Step 1: Understanding the Exercise

We are asked to find the right-handed limit of the function \( \lim_{x \to -8^+} \frac{2x}{x+8} \). This means we need to determine the behavior of the function as \( x \) approaches \( -8 \) from values greater than \( -8 \).

Step 2: Identify the Problem Function

The function given is \( f(x) = \frac{2x}{x+8} \). As \( x \) approaches \( -8 \) from the right, the denominator \( x + 8 \) approaches zero, which suggests a potential vertical asymptote at \( x = -8 \).

Step 3: Analyzing the Sign of the Function

For \( x > -8 \), \( x+8 \) is small positive. Therefore, the expression \( \frac{2x}{x+8} \) will have \( 2x \) as negative because \( x < 0 \) and \( x+8 \) as positive, leading to an overall negative quotient. Thus, \( \frac{2x}{x+8} \rightarrow -\infty \) as \( x \rightarrow -8^{+} \).

Step 4: Conclusion of the Limit Analysis

By evaluating the function's behavior as \( x \rightarrow -8^{+} \), we see that the function approaches negative infinity. This matches the expectation of a vertical asymptote from the right side.

Key Concepts

Right-Hand Limit

A right-hand limit, also known as a right-sided limit, is used to evaluate the behavior of a function as the input approaches a specific point from the right side. In other words, for the function \( f(x) \), when we calculate \( \lim_{x \to a^+} f(x) \), we are interested in understanding what value the function is approaching as \( x \) becomes very close to \( a \) but remains slightly larger than \( a \).

• To find the right-hand limit, identify the target point \( a \) and ensure that \( x > a \).
• Check how the function behaves as \( x \) diminishes towards \( a \), keeping it on the greater side.

For example, in our exercise, the right-hand limit \( \lim_{x \to -8^+} \frac{2x}{x+8} \) involves observing how the function behaves as \( x \) approaches \(-8\) from values greater than \(-8\). By understanding right-hand limits, you are better prepared to analyze the overall behavior of functions at specific points, especially around vertical asymptotes.

Vertical Asymptote

Vertical asymptotes occur at points where a function increases or decreases without bound, meaning it heads off to infinity or negative infinity. These are typically found where the denominator of a rational function approaches zero, causing the function's value to skyrocket.

• Identify where the denominator is zero without the numerator being zero at the same time.
• Examine the behavior of the function as it approaches these points from both sides—left and right.

In our specific case, the function \( f(x) = \frac{2x}{x+8} \) is analyzed around \( x = -8 \). As \( x \rightarrow -8^+ \), the denominator \( (x+8) \) trends towards zero. Since the numerator doesn't approach zero at the same rate, there is a vertical asymptote. Recognizing vertical asymptotes is crucial because they inform us of points of discontinuity, helping us to graph and understand the function's behavior in greater depth.

Function Analysis

Function analysis involves breaking down the function to understand its overall behavior, including identifying limits, asymptotes, and possible discontinuities. This helps in constructing a broad understanding of how the function behaves across its domain.

• Determine the domain of the function by identifying values for which it is undefined.
• Investigate the function’s behavior at particular points or intervals, such as near asymptotes or extrema.

For \( f(x) = \frac{2x}{x+8} \), analyzing the function requires evaluating how \( 2x \) and \( (x+8) \) interact as \( x \) approaches unusual values such as \(-8\). Such analysis can lead to conclusions about the limits, such as a trend towards negative infinity in our calculation. This structured analysis of functions aids in predicting performance patterns and preparing for complex graphing efforts.