Problem 1 - Calculus: Early Transcendentals

Question

Explain in your own words what is meant by the equation \( \displaystyle \lim_{x\to 2} f(x) = 5 \) Is it possible for this statement to be true and yet \( f(2) = 3 \)? Explain.

Step-by-Step Solution

Answer

Yes, the limit can be 5 while \( f(2) \) is 3, as the limit only depends on values near 2, not at 2.

Step 1: Understanding Limits

The notation \( \lim_{x \to 2} f(x) = 5 \) indicates that as \( x \) approaches the value of 2, the function \( f(x) \) approaches the value of 5. Importantly, this statement does not necessarily involve the value of the function \( f \) at \( x = 2 \). It is entirely focused on the behavior of \( f(x) \) as \( x \) gets closer to, but is not necessarily equal to, 2.

Step 2: Examining the Function at the Limit Point

The value of the function \( f(x) \) at the point \( x = 2 \), denoted as \( f(2) \), can potentially be different from the value that \( f(x) \) approaches as \( x \) approaches 2. This means that \( f(2) \) can be any number without affecting the limit, as long as \( f(x) \) near \( x = 2 \) is close to 5.

Step 3: Assessing the Contradiction or Consistency

Since the definition of a limit allows the value of the function at the point of interest to be different from the limit itself, it is completely possible for \( \lim_{x \to 2} f(x) = 5 \) to be true while \( f(2) = 3 \). These statements do not contradict each other because a limit only considers values close to but not always at the point itself.

Key Concepts

Limit of a function

When we talk about the limit of a function, we refer to the value that the function approaches as the input nears a certain point. For example, in the expression \( \lim_{x\to 2} f(x) = 5 \), we are interested in what values \( f(x) \) takes on as \( x \) gets closer and closer to 2. However, it's important to remember that the actual value of the function at 2 isn't part of the limit's definition. This means:

• The function's behavior near the point matters more than its behavior at the point itself.
• It's about observing the trend or pattern the function follows as it gets infinitesimally close to the specified point.

The limit is a fundamental concept because it helps us understand functions and their behaviors even when the function is not explicitly defined at that specific point. This makes limits a crucial tool in calculus and mathematical analysis.

Continuity

Continuity in a function describes a smooth, unbroken path of values for the function over an interval. For a function to be continuous at a point, the following conditions must be satisfied:

• The limit of the function as it approaches the point exists.
• The value of the function at that point is defined.
• The limit as the function approaches the point equals the function’s value at that point.

If any of these conditions fail, the function is not continuous at that point. For example, using \( \lim_{x\to 2} f(x) = 5 \) and \( f(2) = 3 \), the function isn't continuous at \( x = 2 \) because the approaching value (5) doesn't match the value of the function at 2 (which is 3). Understanding continuity is essential as it ensures there are no jumps, breaks, or holes in the function across its domain.

Value of a function at a point

The value of a function at a specific point is simply the output that the function provides when the input equals that specific value. In mathematical terms, for a function \( f(x) \), \( f(c) \) represents the function's value at the point \( c \). Using our earlier example: when \( x = 2 \), \( f(2) = 3 \) tells us that the output of the function at that precise input is 3.

• The specific value of a function at a point can differ from the limit, as we've discussed.
• It's also important to note that sometimes a function might not even be defined at a point, even though its limit exists.

Conceptually, the value of a function at a point gives us exact data for that specific input, whereas the limit provides insight into the function's overall behavior around that input.