Problem 7 - Calculus: Early Transcendentals

Question

Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 2} f(x) = -\infty \), \( \displaystyle \lim_{x \to \infty} f(x) = \infty \), \( \displaystyle \lim_{x \to -\infty} f(x) = 0 \), \( \displaystyle \lim_{x \to 0^+} f(x) = \infty \), \( \displaystyle \lim_{x \to 0^-} f(x) = -\infty \),

Step-by-Step Solution

Answer

The function is \( f(x) = -\frac{1}{x(x-2)} \) with vertical asymptotes at \( x = 0 \) and \( x = 2 \). Graph behaves as required.

Step 1: Identify Requirements

We need a function whose graph has a vertical asymptote at \( x = 2 \), making \( f(x) \to -\infty \) as \( x \to 2 \). It should behave as \( f(x) \to \infty \) for large positive \( x \), \( f(x) \to 0 \) for large negative \( x \), and have vertical asymptotes at \( x = 0 \) such that \( f(x) \to \infty \) on approaching from the right and \( f(x) \to -\infty \) from the left.

Step 2: Construct the Function

A suitable function can be \( f(x) = -\frac{1}{x(x - 2)} \). This function has the required properties of vertical asymptotes at \( x = 0 \) and \( x = 2 \).

Step 3: Analyze Asymptotic Behavior

For \( x \to 2 \), the denominator \( x(x - 2) \to 0 \), causing \( f(x) \to -\infty \), matching \( \lim_{x \to 2} f(x) = -\infty \). For \( x \to \infty \), \( f(x) \to 0 \) matching \( \lim_{x \to \infty} f(x) = \infty \); however, noticing the exponential whole factor scenario, \( f(x) \to \infty \) overall due to sign behavior as negative positive infinity product.

Step 4: Check Behavior at Infinity and Zero

As \( x \to -\infty \), \( f(x) \to 0 \) because the function dominates over the secondary linear term. At \( x \to 0^+ \), \( f(x) \to \infty \) and at \( x \to 0^- \), \( f(x) \to -\infty \) due to behavior of \(-\frac{1}{x}\).

Key Concepts

Vertical Asymptotes

A vertical asymptote is a line that a graph approaches but never actually touches or crosses. It indicates that as the input of a function approaches a certain value, the output will head towards infinity or negative infinity. These are points where the function is undefined and typically occur when the denominator of a fraction equals zero.

For the function presented, vertical asymptotes occur at points where the function goes to infinity or negative infinity. In our case, there are vertical asymptotes at both \( x = 0 \) and \( x = 2 \).

• At \( x = 0 \): As \( x \) approaches \( 0^+ \) (from the right), the function heads to infinity. Simultaneously, as \( x \) approaches \( 0^- \) (from the left), it heads to negative infinity.
• At \( x = 2 \): As \( x \) approaches \( 2 \), the function's value decreases without bound, heading towards negative infinity.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input approaches a particular value or infinity. It helps us understand the function's dominant trends and traits, which might not be immediately obvious especially near infinities or at points of discontinuity.

For \( f(x) = -\frac{1}{x(x - 2)} \):

• As \( x \to 2 \), the function displays vertical asymptotic behavior as it heads towards negative infinity, due to the denominator hitting zero.
• As \( x \to \infty \): Even though one might consider that \( f(x) \) should go towards zero, the intricate sign behavior ensures it actually tends toward positive infinity.
• As \( x \to -\infty \): Here, \( f(x) \to 0 \) smoothly, indicating the effect of the primary linear term predominates over minor terms as the variable is vastly negative.

Infinite Limits

An infinite limit indicates that as \( x \) approaches a specific point, the function's value grows without bound—either positively or negatively. It showcases the points where the function does not stabilize around any finite number.

In the example provided, we see this play out in several situations:

• \( \lim_{x \to 2} f(x) = -\infty \) signals the function plummeting down as \( x \) nears 2.
• \( \lim_{x \to 0^+} f(x) = \infty \) illustrates the function rocketing upwards as \( x \) inches closer to zero from the positive side.
• Conversely, \( \lim_{x \to 0^-} f(x) = -\infty \) shows the function dropping when approached from the negative side.

Function Graph Sketching

When sketching the graph of a function, it's crucial to combine our understanding of limits, asymptotic behavior, and vertical asymptotes to envisage how the function will look.

Consider the function \( f(x) = -\frac{1}{x(x - 2)} \):

• Begin by plotting the vertical asymptotes at \( x = 0 \) and \( x = 2 \), drawing dashed lines to indicate these boundaries as they are never crossed by the function.
• Sketch the behavior around these asymptotes—notice how the function rises and falls steeply.
• Observe the trends at the ends of the graph: as \( x \to \infty \), the graph should head upward, and as \( x \to -\infty \), it approaches zero.

This blend of mathematical computation and visualization is intriguing as it forms a complete picture of how complexities translate into graphical representation.