Question

Explain the meaning of \(\lim _{x \rightarrow-\infty} f(x)=10\)

Short answer

Answer: The limit \(\lim _{x \rightarrow-\infty} f(x)=10\) represents that as the input \(x\) becomes increasingly large in the negative direction (approaches negative infinity), the output, \(f(x)\), approaches the value of \(10\). This limit also indicates the asymptotic behavior of the function, which means it gets closer to the horizontal line \(y=10\) as \(x\rightarrow -\infty\), but never actually reaches it. The concept of limits like this is important in various real-world scenarios, such as physics, biology, medicine, and finance.

Step-by-step solution

Understanding Limits

A limit is a value that a function approaches as the input (x) approaches a certain value. In the given exercise, we are asked to explain the meaning of \(\lim _{x \rightarrow-\infty} f(x)=10\). This means that as the input \(x\) gets closer and closer to negative infinity, the value of the function \(f(x)\) approaches \(10\).

Asymptotic Behavior

The given limit indicates the asymptotic behavior of the function \(f(x)\). This means that when \(x\) becomes increasingly large in the negative direction, the function \(f(x)\) gets closer and closer to the horizontal line \(y=10\). This line, in this case, is called a horizontal asymptote.

Graphical Interpretation

If we look at the graph of the function, we'll see that the function \(f(x)\) will never actually reach the horizontal line \(y=10\) as \(x \rightarrow -\infty\). However, the graph of the function will get closer and closer to the line without ever touching it.

Real-World Examples

Understanding the meaning of \(\lim _{x \rightarrow-\infty} f(x)=10\) is important in various real-world scenarios. For example, in physics, it can be used to describe the behavior of an object as it moves towards a specific point; in biology or medicine, it can be related to the decay or growth of certain quantities; in finance, it can be related to the behavior of stocks or other financial assets over time. In conclusion, the meaning of \(\lim _{x \rightarrow-\infty} f(x)=10\) is that as the input \(x\) gets larger and larger in the negative direction (i.e., approaches negative infinity), the output, \(f(x)\), gets closer and closer to the value of \(10\). This also tells us about the asymptotic behavior of the function as \(x\) approaches negative infinity.

Key concepts

Asymptotic Behavior

When learning calculus, the concept of asymptotic behavior is a fundamental one. What does it mean? Imagine you are walking towards a wall and with each step, you cover just half the distance to the wall. Theoretically, you'll continue to get closer without ever actually reaching the wall. In mathematical terms, your distance to the wall as you keep walking is an example of asymptotic behavior.

In the context of a function, such as the one given in the exercise \( \lim _{x \rightarrow -\infty} f(x)=10 \), asymptotic behavior refers to how the value of the function \(f(x)\) behaves as the input \(x\) becomes very large in the negative direction. The function approaches a certain value — in this case, 10 — but never quite reaches it, no matter how far along the x-axis we travel. This concept is not just a quirk of mathematical theory, but appears in numerous real-world scenarios, like the rate at which an excited atom releases energy as it stabilizes, or the diminishing force of gravity as one moves far away from the source of mass.

Horizontal Asymptote

Tied closely to asymptotic behavior is the concept of a horizontal asymptote. This is a horizontal line that a function approaches as the input either increases without bound or decreases without end. In the exercise, we see that \( \lim _{x \rightarrow -\infty} f(x)=10 \) indicates the function has a horizontal asymptote at \( y=10 \).

It's essential to understand that the horizontal asymptote is not an absolute barrier that the function cannot cross; rather, it's an indication of the function's behavior at infinity. In some cases, a function may cross its horizontal asymptote at finite points before settling close to its level at infinity. To spot a horizontal asymptote on a graph, we look for a line that the curve approaches as we follow it left or right towards infinity. Understanding horizontal asymptotes helps in predicting long-term behavior of growth or decay processes in various fields like economics, pharmacokinetics, and even celestial mechanics.

Graphical Interpretation of Limits

A picture is worth a thousand words, and this is often true when it comes to the graphical interpretation of limits. Seeing a limit expressed graphically can illuminate its behavior in a way that formulas alone cannot. When discussing the limit \( \lim _{x \rightarrow -\infty} f(x)=10 \) graphically, we are essentially looking at what happens to the curve of \(f(x)\) on a graph as we move leftward along the x-axis towards negative infinity.

The value of the function gets ever closer to \(y=10\), but crucially, no matter how far you extend the graph to the left, the curve never actually touches the line \(y=10\). This graphical approach can be particularly useful for visual learners as it conveys the concept of limits in a visual format. It also helps in anticipating the shape and direction of curves when dealing with more complex functions in calculus, enabling a more intuitive understanding of their behavior at different points.