Question

Use a graph to explain the meaning of \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\)

Short answer

Answer: The graph of the limit \(\lim_{x \rightarrow a^{+}} f(x) = -\infty\) represents a function that approaches negative infinity as the value of \(x\) gets closer to the point 'a' from the right side (values greater than 'a'). The graph will have a vertical asymptote located at \(x=a\), and the function \(f(x)\) will continuously decrease as it approaches this asymptote from the right.

Step-by-step solution

Understand the limit notation

The symbol \(\lim_{x \rightarrow a^{+}} f(x) = -\infty\) means that as \(x\) approaches \(a\) from the right (or values greater than \(a\)), the value of the function \(f(x)\) approaches negative infinity.

Identify the point 'a' on the x-axis

Mark a point 'a' on the x-axis. This is the point that \(x\) approaches from the right.

Sketch the function

Draw the function \(f(x)\) such that it has a vertical asymptote at \(x=a\). This means that the function \(f(x)\) will approach negative infinity as \(x\) comes closer to 'a' from the right side.

Arrow to indicate the limit

Add an arrow pointing down from the function, right before it reaches the vertical asymptote at \(x=a\). This arrow signifies that the function is approaching negative infinity.

Label the graph

Label the graph with the limit expression, \(\lim_{x \rightarrow a^{+}} f(x) = -\infty\), next to the arrow. This will help visualize the concept of the limit and indicate the direction that the function heads as \(x\) approaches 'a' from the right. In conclusion, the graph effectively illustrates the meaning of \(\lim_{x \rightarrow a^{+}} f(x) = -\infty\), which states that as \(x\) gets closer to 'a' from the right side, the value of \(f(x)\) keeps decreasing and approaching negative infinity.

Key concepts

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a particular point or infinity. When examining limits at infinity, or as a function approaches a certain x-value, the term "asymptotic" helps describe the tendency of the graph. As the graph approaches a line, axis, or point without actually touching it, it provides information about the function's end behavior or "approach to infinity or zero."

In the context of the problem, you might see that the function starts sky-high or deep below, and heads towards negative infinity as a point is neared from one side. Most importantly, an asymptote guides how the function behaves in this transition phase.

• Horizontal asymptotes focus on end behavior as x goes to positive or negative infinity.
• Vertical asymptotes, as in our problem, describe behavior near a particular value.

Recognizing asymptotic behavior is key to understanding limits, especially at infinity, since it shows whether a function is increasing, decreasing, or leveling out as it approaches that significant point.

Vertical Asymptote

A vertical asymptote is a line on the graph where the function becomes unbounded, meaning the values of the function increase or decrease indefinitely as they approach this line. In simpler terms, the graph shoots up or down very steeply and approaches infinity or negative infinity. This behavior often occurs because of division by zero or logarithmic functions nearing undefined points.

For the given exercise with a limit going to negative infinity as x approaches 'a' from the right, a vertical asymptote is evident at x = a. This line acts as a barrier where the function cannot touch, yet it demonstrates how the function will behave close to this line.

Vertical asymptotes are vital as they highlight where the function changes significantly and essentially where the function doesn't exist in its normal form. It helps graphically interpret where the function will "shoot off" to infinity or negative infinity.

One-sided Limits

One-sided limits focus on how a function behaves as it approaches a specific point from one direction, either from the left (\( x \rightarrow a^{-} \)) or from the right (\( x \rightarrow a^{+} \)). These kinds of limits are especially useful when dealing with functions that aren't continuous across the whole number line or have different behaviors on either side of a point.

In the context of the exercise, \( \lim_{x \rightarrow a^{+}} f(x) = -\infty \) tells us that as x moves towards the value 'a' from numbers greater than 'a' (or the right), the function value decreases without bound towards negative infinity.

• It indicates the direction of approach—for instance, from the right (\( a^{+} \)).
• Gives insight into how function behavior changes around 'a' on that specific side.
• Useful in piecewise functions where behavior can change sharply between sections.

By examining one-sided limits, we accurately describe and graphically show how the function behaves near discontinuities or boundaries, helping to predict or illustrate potential asymptotic behavior.