Question

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

Short answer

Based on the given conditions and the step-by-step solution, a possible graph of the function \(f\) has a vertical asymptote at \(x=0\) and horizontal asymptotes at \(y=1\) and \(y=-2\). For \(0<x<\infty\), the function approaches the vertical asymptote as \(x \rightarrow 0^+\) and the horizontal asymptote at \(y=1\) as \(x \rightarrow \infty\). For \(-\infty<x<0\), the function approaches the vertical asymptote as \(x \rightarrow 0^-\) and the horizontal asymptote at \(y=-2\) as \(x \rightarrow -\infty\).

Step-by-step solution

Identify the vertical asymptote

Since \(\lim_{x \rightarrow 0^{+}} f(x) = \infty\) and \(\lim_{x \rightarrow 0^{-}} f(x) = -\infty\), there must be a vertical asymptote at \(x=0\).

Identify the horizontal asymptotes

The limits \(\lim_{x \rightarrow \infty} f(x)=1\) and \(\lim_{x \rightarrow -\infty} f(x)=-2\) imply there are horizontal asymptotes at \(y=1\) and \(y=-2\).

Break down each condition

1. \(\lim_{x \rightarrow 0^{+}} f(x)=\infty\): As \(x\) approaches \(0\) from the right, the function \(f(x)\) increases towards infinity. 2. \(\lim_{x \rightarrow 0^{-}} f(x)=-\infty\): As \(x\) approaches \(0\) from the left, the function \(f(x)\) decreases towards negative infinity. 3. \(\lim_{x \rightarrow \infty} f(x)=1\): As \(x\) approaches infinity, the function \(f(x)\) approaches the horizontal asymptote of \(y=1\). 4. \(\lim_{x \rightarrow-\infty} f(x)=-2\): As \(x\) approaches negative infinity, the function \(f(x)\) approaches the horizontal asymptote of \(y=-2\).

Sketch the graph

Now we are ready to sketch the graph of the function \(f\). 1. Draw a dotted vertical line at \(x=0\) (the vertical asymptote) 2. Draw dotted horizontal lines at \(y=1\) and \(y=-2\) (the horizontal asymptotes) 3. Draw a curve for \(0<x<\infty\) where the curve approaches the vertical asymptote as \(x \rightarrow 0^+\) and the horizontal asymptote at \(y=1\) as \(x \rightarrow \infty\). 4. Draw a curve for \(-\infty<x<0\) where the curve approaches the vertical asymptote as \(x \rightarrow 0^-\) and the horizontal asymptote at \(y=-2\) as \(x \rightarrow -\infty\). The resulting sketch will show a graph that satisfies all of the given conditions.

Key concepts

Asymptotes

In calculus, an asymptote is a line that the graph of a function approaches as the inputs reach infinity or a specific value, but never actually touches. Asymptotes are important in sketching graphs because they provide critical insight into the behavior of the function at the extremes. There are two main types of asymptotes: horizontal and vertical.

Vertical Asymptotes

Vertical asymptotes occur at specific input values for which the function heads off to infinity. They signify that the function is undefined at these input values. For instance, in the given problem, we identify a vertical asymptote at x=0 based on the limits \(lim_{x \rightarrow 0^{+}} f(x) = \infty\) and \(lim_{x \rightarrow 0^{-}} f(x) = -\infty\). This tells us that the function behaves wildly—skyrocketing upwards (toward positive infinity) as we approach 0 from the right and plummeting downwards (toward negative infinity) when we approach from the left.

Horizontal Asymptotes

Horizontal asymptotes, on the other hand, give us information about how the function behaves as the input gets very large or very small—in other words, as x approaches positive or negative infinity. In our exercise, \(lim_{x \rightarrow \infty} f(x) = 1\) and \(lim_{x \rightarrow -\infty} f(x) = -2\) indicate that the graph will flatten out and get closer and closer to the lines y=1 and y=-2 respectively, but not actually touch them.

Limits of Functions

The limits of functions are foundational to understanding calculus, especially when identifying the behavior of a function as inputs approach certain values. Computing limits allows us to figure out what's happening to the function at points where it might not be well-defined or where it could behave unexpectedly. For sketching graphs, knowledge of limits is especially important as it directly influences the shape and direction of the graph near potential points of discontinuity, such as asymptotes.

In the exercise, the limits clearly define the function's behavior at key points. Here are the effects of each of the given limits:

• \(lim_{x \rightarrow 0^{+}} f(x) = \infty\) means that as x approaches zero from the right, f(x) increases without bound.
• \(lim_{x \rightarrow 0^{-}} f(x) = -\infty\) signifies that as x approaches zero from the left, f(x) decreases without bound.
• With \(lim_{x \rightarrow \infty} f(x) = 1\), we anticipate the function leveling out as x increases.
• \(lim_{x \rightarrow -\infty} f(x) = -2\) indicates that the function levels out in the opposite direction, as x decreases.

These limits do not tell us the exact values of the function at these points, but they do tell us about the trends the function follows as it moves towards those points, which is crucial for graphing.

Graphing Functions

The skill of graphing functions translates the behavior of functions, often described algebraically or numerically, into a visual format. This visual representation is invaluable for communicating complex mathematical concepts effectively. By utilizing the limits and identifying asymptotes, as seen in our exercise, we can outline the overall shape and path of a function's graph even without plotting individual points.

When sketching a graph, it's important to remember the following steps:

• Identify asymptotes as guidelines for where the graph cannot pass through.
• Use limits to determine the end behavior of the graph near these asymptotes and at infinity.
• Combine this information to draw a rough outline of the graph.

In our problem, the function swings upward as it nears zero from the right and downward as it nears zero from the left, while approaching the horizontal lines y=1 and y=-2 as x moves positively and negatively toward infinity, respectively. These behaviors give us enough information to sketch an accurate graph of the function. Always remember that the more information you have (through limits, asymptotes, and other properties), the more accurate your sketched graph will be.