Question

$$\text { Explain the meaning of } \lim _{x \rightarrow a^{+}} f(x)=L$$

Short answer

Answer: The notation "lim_{x → a^+} f(x) = L" represents the limit of a function f(x) as x approaches 'a' from the right side. It denotes the value that the function f(x) approaches (L) as x gets infinitely close to 'a' from the right, i.e., x > a. This concept helps to describe the behavior of a function near a specific point and determine its continuity. The one-sided limit indicates how the function behaves only for values greater than 'a', and for a limit to exist at x = a, the one-sided limits from both the left and the right must be equal.

Step-by-step solution

Understanding the notation

The notation "lim_{x → a^+} f(x) = L" represents the limit of a function f(x) as x approaches 'a' from the right. The '+' sign in 'a^+' indicates that we are only considering values of x on the right side of 'a', i.e., x > a. The limit 'L' is the value f(x) approaches as x gets infinitely close to 'a' from the right side.

Limit behavior as x approaches a

To better understand the limit of a function as x approaches 'a', imagine the function f(x) as a curve on a graph. When x gets very close to 'a', the function approaches some value (L). The function does not need necessarily to be defined at x = a. If the function approaches the same value from both the left and right of 'a', then the limit of the function f(x) as x approaches 'a' exists, and the value it approaches is called L.

Limit example

Consider the following function f(x) = (x^2-1)/(x-1). If we want to find lim_{x → 1^+} f(x), we are looking for the value that f(x) approaches as x gets very close to '1' from the right side. Note that x = 1 is not in the domain of the function, as it yields a division by zero. However, by factoring f(x), we can rewrite it as f(x) = (x+1) for x ≠ 1. Now it is clear that for x > 1, lim_{x → 1^+} f(x) = 2, since f(x) approaches '2' as x gets infinitely close to '1' from the right.

One-sided limits and graph behavior

One-sided limits, such as the lim_{x → a^+} f(x), describe the behavior of a function only considering a specific direction, in this case, x > a. By inspecting the graph of f(x) for values x > a, we can determine whether the function approaches a specific value L as x approaches 'a' from the right. If it does, then we can state that lim_{x → a^+} f(x) = L. Keep in mind that for a limit to exist at x = a, the one-sided limits coming from both the left (lim_{x → a^-} f(x)) and the right (lim_{x → a^+} f(x)) must be equal.

Key concepts

Limits of a Function

When we speak of the limits of a function, we are addressing a fundamental concept in calculus that deals with the behavior of that function as the input values approach a certain point. Limits help us understand what value a function heads towards, which is not always the same as the function's actual value at that point. They play a crucial role in defining continuity, derivatives, and integral calculus, essentially providing insight into the function's tendency near a specific input, even if the function is not clearly defined there.
For example, consider the function defined by the rule that assigns to each input its square minus two. If we want to know what happens as we get close to the input three, we look at the outputs for inputs like 2.9, 2.99, 2.999, and so on. We see that the outputs get closer and closer to seven, and we say that the limit of the function as the input approaches three is seven, even if the function at exactly three is something different.

Limit Notation

Limit notation is a standardized way of expressing the concept of limits in mathematics. The notation \( \lim_{x \rightarrow a} f(x) = L \) tells us that as the variable \( x \) approaches the value \( a \) from either side on the number line, the function \( f(x) \) approaches the value \( L \) as its limit. This expression does not necessarily mean that \( f(a) = L \) or even that \( f(x) \) is defined when \( x = a \)—it speaks purely about the behavior of the function near \( a \) without concerning itself with the exact value at \( a \).
In our example above, when we're looking at the input square minus two as it gets close to three, and its output gets close to seven, we use the notation \( \lim_{x \rightarrow 3} (x^2 - 2) = 7 \) to express this situation. This handy shorthand provides a clear and concise way to communicate complex mathematical ideas.

Limit Behavior

Understanding limit behavior involves exploring how a function acts as its input gets infinitely close to a particular value. It's like peering into the function's trend around that input point without actually reaching it. The action of 'approaching' reveals if the function behaves nicely, heading towards a single, well-defined output, or if it does something erratic, like shooting off to infinity or oscillating wildly.
In our given function \( f(x) = \frac{x^2-1}{x-1} \) as an example, near \( x = 1 \) the function simplifies to \( x+1 \) for values of x other than 1. As we approach 1 from either side, the outputs hover around the value 2, indicating that the function 'behaves' by trending towards 2.

Approaching from the Right

Approaching a value from the right, notated as \( \lim_{x \rightarrow a^{+}} f(x) \) in calculus, specifically examines the limit behavior of a function as the input approaches a given value, \( a \) from values greater than \( a \)—in other words, from the right side on a number line. This 'one-sided limit' can be distinct from the limit as it approaches from the left (notated as \( \lim_{x \rightarrow a^{-}} \) ) and offers insight into the function's behavior solely from one direction.
It is like watching a hiker, representing the function, journey towards a mountain peak, which is the value \( a \). If we only explore the hiker's approach from the eastern trail, we get an understanding of the climb solely from that single approach. For example, when we considered the limit of \( \frac{x^2-1}{x-1} \) as \( x \) approaches 1 from the right, we only looked at values like 1.1, 1.01, and 1.001 to understand the function’s trend, which led us to limiting value 2.