Problem 2 - Calculus: Early Transcendentals

Question

Explain what it means to say that \( \displaystyle \lim_{x\to1^-}f(x) = 3 \) and \( \displaystyle \lim_{x\to1^+}f(x) = 7 \) In this situation is it possible that \( \displaystyle \lim_{x\to1}f(x) \) exists? Explain.

Step-by-Step Solution

Answer

The limit \( \lim_{x\to1}f(x) \) does not exist because the left-hand and right-hand limits are not equal.

Step 1: Understanding Left-Hand Limit

The notation \( \lim_{x\to1^-}f(x) = 3 \) implies that as \( x \) approaches 1 from the left (values less than 1), the function \( f(x) \) approaches a limit of 3. This describes the behavior of the function for values of \( x \) that are slightly less than 1.

Step 2: Understanding Right-Hand Limit

The notation \( \lim_{x\to1^+}f(x) = 7 \) indicates that as \( x \) approaches 1 from the right (values greater than 1), the function \( f(x) \) approaches a limit of 7. This describes the behavior of the function for values of \( x \) that are slightly greater than 1.

Step 3: Checking the Existence of the Limit

For \( \lim_{x\to1}f(x) \) to exist, both the left-hand limit and the right-hand limit as \( x \to 1 \) must be equal. Since \( \lim_{x\to1^-}f(x) = 3 \) and \( \lim_{x\to1^+}f(x) = 7 \), these limits are not equal, meaning that \( \lim_{x\to1}f(x) \) does not exist.

Key Concepts

Understanding the Left-Hand Limit

When we talk about the left-hand limit, we mean how a function behaves as the input values approach a certain number from the smaller side. This is symbolized as the minus sign in the notation \( \lim_{x \to c^-} f(x) \). In our example, \( \lim_{x \to 1^-} f(x) = 3 \) tells us that as we get very close to 1 from numbers less than 1, the output of the function \( f(x) \) gets closer and closer to 3. This is like approaching the number 1 by moving from 0.9 toward 1 while seeing how the function behaves. You can think of it like watching a value sneak up on 1 from the left-hand side. The closer you get to 1 from this side, the more the value of \( f(x) \) approaches 3.

• It describes behavior as \( x \) moves closer to 1 from the left or smaller numbers.
• Important for understanding changes in behavior near specific points on the graph.

Knowing the left-hand limit is crucial because it helps us predict the function's behavior near a particular point from one direction.

Understanding the Right-Hand Limit

Just like we have the left-hand limit, the right-hand limit helps us understand what happens as we approach a certain point from the numbers bigger than it. It's represented by the plus sign in the limit notation \( \lim_{x \to c^+} f(x) \). Imagine you have \( \lim_{x \to 1^+} f(x) = 7 \); this tells us that as \( x \) approaches 1 from values slightly larger than 1, the function approaches a value of 7. Heading toward 1 from the right means coming down from numbers like 1.1 to 1.01 and seeing how the function behaves as we get closer to 1.

• It describes behavior as \( x \) moves closer to 1 from the right, or larger numbers.
• Helps determine function behaviors at key points.

Both left-hand and right-hand limits offer a thorough look at the function's behavior near critical points, showing us how it approaches those points from different sides.

Understanding the Existence of a Limit

For a limit to truly "exist" at a certain point, both the left-hand limit and right-hand limit must agree. This means they must approach the same number as they close in on a specific point. In simpler terms, if we say \( \lim_{x \to 1} f(x) \) exists, both \( \lim_{x \to 1^-} f(x) \) and \( \lim_{x \to 1^+} f(x) \) must be equal. In our exercise, we see that \( \lim_{x \to 1^-} f(x) = 3 \) and \( \lim_{x \to 1^+} f(x) = 7 \), meaning as we approach 1 from both sides, we get different values (3 and 7).

• A single, unified limit requires both side limits to match.
• Without equality, the limit does not exist at that point.

If both sides don't match, the function doesn't have a singular value to approach. Thus, \( \lim_{x \to 1} f(x) \) does not exist since these values are unequal. Observing this helps understand parts of a function's graph that might have jumps or stops and why a single, clean graphed line isn't possible at that spot.