Question

Sketch a possible graph of a function \(f\) that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$\lim _{x \rightarrow 0^{+}} f(x)=\infty, \lim _{x \rightarrow 0^{-}} f(x)=-\infty, \lim _{x \rightarrow \infty} f(x)=1$$, $$\lim _{x \rightarrow-\infty} f(x)=-2$$

Short answer

Question: Sketch a graph of a function f(x) that satisfies the following conditions: 1. As x approaches 0 from the positive side, f(x) goes to infinity. 2. As x approaches 0 from the negative side, f(x) goes to negative infinity. 3. As x approaches infinity, f(x) goes to 1. 4. As x approaches negative infinity, f(x) goes to -2. Answer: To sketch the graph of the function f(x), begin by drawing a vertical asymptote at x=0. Next, draw two horizontal asymptotes at y=1 (as x approaches infinity) and y=-2 (as x approaches negative infinity). The graph should approach the vertical asymptote and go upward to infinity as x approaches 0 from the positive side, and downward to negative infinity as x approaches 0 from the negative side. As x approaches infinity, the graph should get closer to the horizontal asymptote at y=1, and as x approaches negative infinity, it should get closer to the horizontal asymptote at y=-2. Once these conditions are satisfied, the graph of the function f(x) will meet all given conditions.

Step-by-step solution

Identify Vertical Asymptote

As \(x\) approaches 0 from both sides, \(f(x)\) goes to positive and negative infinity. Therefore, there is a vertical asymptote at \(x=0\).

Identify Horizontal Asymptotes

As \(x\) approaches infinity, \(f(x)\) approaches the value 1. This means that there is a horizontal asymptote at \(y=1\) as \(x\) goes to infinity. Similarly, as \(x\) approaches negative infinity, \(f(x)\) approaches the value -2. This means that there is another horizontal asymptote at \(y=-2\) as \(x\) goes to negative infinity.

Sketch the Graph

Start by drawing the vertical asymptote at \(x=0\). This is a vertical dashed line. Next, draw the two horizontal asymptotes. At \(y = 1\) (as x goes to infinity) and at \(y = -2\) (as x goes to negative infinity). These are horizontal dashed lines. Now, keeping the given conditions in mind, sketch the graph. 1. As \(x\) approaches 0 from the positive side, the graph of the function should approach the vertical asymptote and go upward to infinity. 2. As \(x\) approaches 0 from the negative side, the graph of the function should approach the vertical asymptote and go downward to negative infinity. 3. When moving towards positive infinity in the x-axis, the graph of the function should get closer to the horizontal asymptote at \(y=1\). It should never cross the asymptote but get arbitrarily close to it as x gets larger. 4. When moving towards negative infinity in the x-axis, the function's graph should get closer to the horizontal asymptote at \(y=-2\). It should never cross the asymptote but get arbitrarily close to it as x gets smaller. Once these conditions are satisfied, the graph of the function \(f(x)\) will meet all the given conditions.

Key concepts

Graph Sketching

Creating a sketch of a function involves visually representing key behaviors, such as how a function increases or decreases, where it approaches infinity, and the asymptotes present in the graph. This helps in visualizing complex functions and understanding their behavior without needing precise calculations at every point.

When sketching a graph, you start by noting vertical and horizontal asymptotes because they guide the general flow of the function. An asymptote is a line that the graph approaches but never touches, acting like an invisible boundary. These lines are sometimes drawn as dashed to differentiate them from the solid lines representing the functions themselves.

To effectively sketch a graph, follow these steps:

• Identify and draw the asymptotes on the grid.
• Determine the overall trend of the graph between the asymptotes.
• Be aware of any points where the curve may change in direction or slope.

This process gives a clear outline of the function’s behavior, making it easier to analyze or predict specific values.

Limits

Limits help us understand the behavior of a function as it approaches a particular point from either side. They are the foundation of calculus and are crucial for understanding continuity, derivatives, and integrals.

Consider the exercise's conditions: - As x approaches 0 from the positive side, the function heads towards infinity ( \( \lim _{x \rightarrow 0^{+}} f(x)=\infty \)). - Approaching 0 from the negative gives a limit of negative infinity ( \( \lim _{x \rightarrow 0^{-}} f(x)=-\infty \)).

These indicate that as x is close to 0, but not actually zero, the value of the function grows without bounds either positively or negatively. It reveals a vertical asymptote at x = 0, as discussed in the next sections.

Additionally, limits describe the behavior as x becomes very large or very small (approaching positive or negative infinity), leading us to horizontal asymptotes. These asymptotes provide insight into how the values stabilize or balance out over an extensive range.

Horizontal Asymptotes

A horizontal asymptote reflects the value that a function approaches as the independent variable (x) moves towards infinity or negative infinity. Horizontal asymptotes describe the long-term behavior of functions and aid in predicting their behavior for extreme values.

In the given exercise, the limit states:- As x approaches positive infinity, f(x) nears 1 ( \( \lim _{x \rightarrow \infty} f(x)=1 \)).- As x approaches negative infinity, f(x) approaches -2 ( \( \lim _{x \rightarrow -\infty} f(x)=-2 \)).

These conditions establish the horizontal asymptotes y = 1 and y = -2. Essentially, as x stretches far into the positives or negatives, the function doesn't surpass these lines but gets exceedingly close, leveling off as if the curve "hugs" the asymptote. Whether the function creeps close from above or below depends on the specific function and its starting point.

Vertical Asymptotes

A vertical asymptote is a vertical line on a graph where the function f(x) increases or decreases without bound as x approaches a particular value. This suggests a point of discontinuity at which the function is undefined.

In the context of the exercise:
- As x approaches 0 from either side, f(x) heads towards either positive or negative infinity ( \( \lim _{x \rightarrow 0^{+}} f(x)=\infty \) and \( \lim _{x \rightarrow 0^{-}} f(x)=-\infty \)).

This behavior signifies a vertical asymptote at x = 0. In practical terms, as one moves closer to zero on the graph from either direction, the value of f(x) escalates steeply, making a "cliff-like" appearance on the graph.

Vertical asymptotes are often accompanied by a change in sign of f(x), thus the function might rise or fall abruptly as it approaches the asymptote. When sketching, mark these lines clearly with dashed lines, indicating the barrier that the function approaches but never intersects.