Problem 1 - APEX Calculus

Question

In your own words, what does it mean to "find the limit of \(f(x)\) as \(x\) approaches \(3^{\prime \prime} ?\)

Step-by-Step Solution

Answer

Find the value that \( f(x) \) approaches as \( x \) gets closer to 3 from both sides.

Step 1: Understanding Limits

Finding the limit of a function involves determining the value that the function approaches as the input approaches a given number or point. In this case, we are looking at how the function behaves as the input, denoted as \( x \), gets closer and closer to 3.

Step 2: Identifying the Function Behavior

To find the limit of \( f(x) \) as \( x \) approaches 3, consider how the values of the function \( f(x) \) behave when \( x \) takes values increasingly close to 3, from both the left (values less than 3) and the right (values greater than 3).

Step 3: Evaluating Left-Hand and Right-Hand Limits

Calculate \( \lim_{{x \to 3^-}} f(x) \), which is the limit from the left, and \( \lim_{{x \to 3^+}} f(x) \), which is the limit from the right. If both limits exist and are equal, this common value is the limit of \( f(x) \) as \( x \) approaches 3.

Step 4: Concluding the Limit

If the left-hand limit and the right-hand limit at \( x = 3 \) are equal, then the limit of \( f(x) \) as \( x \) approaches 3 is equal to this common value. Otherwise, the limit does not exist.

Key Concepts

function behavior

When studying limits in calculus, understanding the behavior of a function is vital. This behavior indicates how the function's values change as the input approaches a certain point, like 3 in our example. Imagine a road that leads to a bridge: knowing the behavior is like observing how cars (or the function’s values) move and adjust as they get closer to the start of the bridge.

• If the cars slow down smoothly, you know there's likely a slow-moving traffic up ahead.
• However, if cars change lanes or stop abruptly, the road conditions may include construction or an obstacle.

Similarly, as we approach a specific point in a function, such as when we get closer and closer to the x-value of 3, the behavior is seen by noting how the values of the function adapt. Thus, understanding function behavior is crucial for defining whether and what kind of limits exist.

left-hand limit

The concept of a left-hand limit helps us understand how a function behaves as we approach a particular point from the left side, or from values less than that point. In more formal mathematical terms, the left-hand limit of a function as x approaches 3, written as \( \lim_{{x \to 3^-}} f(x) \), is the value that \( f(x) \) tends towards as x approaches 3 from the left or from smaller numbers.

• This can be visualized like walking towards a door from a hallway that is on your left.
• Just as you survey what's in front of you as you approach the door from the left side, the left-hand limit observes the behavior of the function’s values as we approach 3 from the left.

Calculating the left-hand limit involves checking the function's values at increasingly smaller intervals leading up to 3. This provides a picture of the function's trend and helps in deciding whether the overall limit exists at that point.

right-hand limit

The right-hand limit complements the left-hand limit by focusing on how a function behaves as we approach a particular point from the right side, or from values greater than that point. In our example, the right-hand limit of a function as x approaches 3, symbolized as \( \lim_{{x \to 3^+}} f(x) \), evaluates what \( f(x) \) approaches as x gets closer to 3 from the right, or from larger numbers.

• This is akin to looking through a window from right to left; you notice what's changing as you move rightward towards your target.
• Step by step, you examine each point at a slightly lower number than where you started.

We calculate the right-hand limit by analyzing the values of the function as they come coming closer to 3 from the right side. This helps in building a comprehensive understanding of the limit, confirming if both left and right perspectives meet at the same point, thus establishing if the limit truly exists.