Problem 71 - Thomas Calculus

Question

In Exercises \(69-72,\) sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required- -just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$f(0)=0, \lim _{x \rightarrow \pm \infty} f(x)=0, \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow-1^{+}} f(x)=\infty,\\\\{\lim _{x \rightarrow 1^{+}} f(x)=-\infty,\text{and}\lim _{x \rightarrow-1^{-}} f(x)=-\infty}$$

Step-by-Step Solution

Answer

Graph passes through (0,0) with vertical asymptotes at x=1 and x=-1.

Step 1: Understand the Conditions

Firstly, understand each condition stated in the exercise. These conditions describe the behavior of the graph of the function around specific points and at infinity.

Step 2: Plot Key Points

Plot the point \((0, 0)\) on the graph since \(f(0)=0\). This is a point through which the graph must pass.

Step 3: Consider Limits at Infinity

The limits \(\lim _{x \rightarrow \pm \infty} f(x)=0\) imply that the graph approaches the x-axis as \(x\) goes to positive or negative infinity.

Step 4: Analyze Asymptotic Behavior Near Given Points

Identify that \(x = 1\) and \(x = -1\) are vertical asymptotes based on the given limits: \(\lim _{x \rightarrow 1^{-}} f(x) = \lim _{x \rightarrow -1^{+}} f(x) = \infty\) and \(\lim _{x \rightarrow 1^{+}} f(x) = \lim _{x \rightarrow -1^{-}} f(x) = -\infty\). This means the graph goes to infinity as \(x\) approaches 1 from the left side and approaches from the right and vice versa at \(-1\).

Step 5: Sketch the Graph

Sketch the graph with vertical asymptotes at \(x = -1\) and \(x = 1\). The graph should rise to \(+\infty\) as \(x\) approaches 1 from the left and -1 from the right, and fall to \(-\infty\) as \(x\) approaches 1 from the right and -1 from the left. Ensure the graph passes through \((0, 0)\) and approaches the x-axis for large positive and negative \(x\).

Step 6: Final Review

Check that the graph satisfies all given behaviors and is appropriately labeled with the x-axis and y-axis. Adjust the sketch if necessary to better reflect the described conditions.

Key Concepts

Asymptotic Behavior

Understanding asymptotic behavior is crucial when analyzing the shape and direction of graphs in mathematics. In simple terms, asymptotic behavior refers to the way a function behaves as its input, usually represented as \(x\), becomes very large or very small.

• Imagine a graph that flattens but never quite touches a line, this line is often called an "asymptote." As the value of \(x\) extends to infinity, the function values approach this line without actually reaching it.
• To visualize: if you move towards the ends of the x-axis on the function's graph, the path of the graph might start following the direction of a horizontal, vertical, or slanting line, known as asymptotic lines.

In the current exercise, asymptotic behavior is seen when the graph approaches the x-axis as \(x\) approaches positive or negative infinity. This results in the behavior of the graph flattening out as it moves away from the center.

Limits at Infinity

Limits at infinity provide insights into what happens to function values as the input becomes extremely large or extremely small. They help us understand the overall behavior of functions in the far ends of the graph.

• When we say \( \lim_{x \rightarrow \pm \infty} f(x) = L \), it means as \(x\) moves either to positive or negative infinity, the function \(f(x)\) approaches a particular value \(L\).
• In many cases, especially for rational functions, limits at infinity can indicate horizontal asymptotes.

In the exercise: \( \lim _{x \rightarrow \pm \infty} f(x) = 0 \) signifies the graph of the function approaches the x-axis when moving towards both positive and negative infinity.

Vertical Asymptotes

A vertical asymptote is a vertical line that a graph approaches but never touches or crosses. Understanding vertical asymptotes is essential for identifying where functions have infinite discontinuities.

• Vertical asymptotes often occur when a function’s denominator becomes zero, leading to values of \(f(x)\) shooting up to positive or negative infinity.
• On a graph, these are seen as lines that the curve gets infinitely close to but never actually meets.

The original exercise points out vertical asymptotes at \(x = 1\) and \(x = -1\). This means that as \(x\) approaches 1 from the left, the graph shoots up to infinity, and from the right, it drops down to negative infinity, showing the discontinuous nature of the function at these points.