Question

In Exercises 1 and \(2,\) describe in your own words what the statement means. $$ \lim _{x \rightarrow-\infty} f(x)=2 $$

Short answer

The given statement means that as 'x' tends towards negative infinity, the function 'f(x)' approaches the value '2'. Thus, no matter how far to the left we move along the x-axis (into the negative direction), the function appears to get closer to '2' but never exactly reaches it.

Step-by-step solution

Understand the Concept of Limit

The concept of limit in calculus is used to describe a value that a function approaches as the input (or variable) approaches a certain value. It is represented as \( \lim _{x \rightarrow a} f(x)=L \), where 'L' is the limit and 'a' is the value that 'x' approaches.

Apply this Understanding to the Given Statement

In the provided statement \( \lim _{x \rightarrow-\infty} f(x)=2 \), it's saying that as 'x' moves further and further into the negative direction (towards negative infinity), the function 'f(x)' gets closer and closer to the value '2'.

Make a Concise Statement

We could concisely express this by saying, 'As 'x' tends towards negative infinity, the function 'f(x)' approaches the value '2'.'

Key concepts

Understanding the Limit Concept

The limit concept in calculus is a fundamental building block for understanding how functions behave. At its core, a limit describes the behavior of a function as the input value gets closer to a particular point, regardless of whether the function actually reaches that point or not.

For instance, when you see an expression like \( \lim _{x \rightarrow a} f(x)=L \), it's asserting that as \( x \) incrementally moves closer to \( a \), \( f(x) \) gets increasingly close to the limit \( L \). It's important to note that \( x \) doesn't need to actually reach \( a \), and \( f(x) \) doesn't need to equal \( L \) for the limit to exist. This is about the trend as \( x \) approaches \( a \), not the specific values at \( x = a \).

The concept is also crucial when dealing with infinite limits or when a function grows without bound. Limits help us mathematically articulate the idea of 'approaching' a value, making precise what might otherwise be a vague notion.

Analyzing Function Behavior

Function behavior looks at how a function acts near a certain point or as it progresses along the x-axis. This is especially intriguing when examining a function’s behavior nearby or at a boundary of its domain, or as it extends towards infinity.

A useful way to visualize this is by considering a graph of the function. As you trace the curve with your eyes, moving along the x-axis, you can watch how the y-value (the output) evolves. Does the graph flatten out, indicating that the function is approaching a horizontal asymptote, does it oscillate, or does it veer off towards infinity?

With the expression \( \lim _{x \rightarrow -\infty} f(x)=2 \), the behavior under scrutiny is as \( x \) becomes very large negatively. Here, our story isn't about the actual values that \( f(x) \) takes on, but about its trend. We say that the y-values 'settle down' to 2, creating a visual image of \( f(x) \) getting indefinitely closer to the horizontal line \( y=2 \) as we look further and further left on the graph.

Approaching Infinity: Grasping Infinite Limits

When we talk about approaching infinity, we are extending the limit concept into the vast, unbounded scales. In calculus, infinity isn't a number you can reach or measure; it's more of an idea, representing something that grows without end.

In the notation \( \lim _{x \rightarrow -\infty} f(x)=L \), the '-\infty' symbolizes that the x-values are decreasing without limit. It's different from ordinary limits where \( x \) approaches a finite number, because here, it's about a journey with no final destination.

This type of limit can assist in understanding long-term behavior of functions. If a function approaches a specific value (like 2 in our example) as \( x \) heads to negative infinity, it suggests a predictable pattern or trend as you move leftward along the x-axis. In other words, no matter how far back you go, the values of \( f(x) \) remain close to the limit value, providing stability in the function's long-range behavior.