Finding Limits: A Step-by-Step Guide

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Hey guys, let's dive into a classic calculus problem! We're going to figure out the limit of a piecewise function. Don't worry, it sounds scarier than it is. We'll break it down into easy steps, and you'll be a limit-finding pro in no time! So, let's get started. Understanding limits is fundamental to calculus. It helps us understand the behavior of functions near certain points, even if the function isn't defined at that exact point. This concept is used in a wide range of applications, from physics to economics, so getting a solid grasp of it is super important.

The Problem Explained

Alright, let's look at the function we have. We're dealing with a piecewise function, which means it's defined differently depending on the value of x. The function f(x) is defined as follows: f(x) = x² - 3 when x is less than 2, and f(x) = (3x + 2) / 4 when x is greater than or equal to 2. Our mission, should we choose to accept it, is to find the limit of f(x) as x approaches 2. In other words, we need to determine what value f(x) gets closer and closer to as x gets closer and closer to 2. This is like zooming in on the graph of the function around x = 2.

Now, the interesting part about piecewise functions is that we have to consider what happens to the function from both sides of the point in question. This means we need to look at the left-hand limit (as x approaches 2 from values less than 2) and the right-hand limit (as x approaches 2 from values greater than 2). If both of these limits exist and are equal, then the overall limit exists, and that’s our answer! We can break this problem down by using the left-hand limit and the right-hand limit. If they are equal, we can assume that the limit exists and we can find the answer.

This approach ensures that we capture the complete behavior of the function near the point in question. Understanding this is a core concept in calculus and is a great way to boost our knowledge. This is a good way to explore the behavior of the function. This helps us determine if the function has a well-defined value, or whether it has a jump or discontinuity at a specific point. Grasping this understanding is crucial. Let's go ahead and see how to determine the value for this function and see if the limit exists.

Step-by-Step Solution

To solve this, we'll follow these steps:

  1. Calculate the Left-Hand Limit: We'll evaluate f(x) as x approaches 2 from the left (i.e., x < 2). In this case, we use the first part of the piecewise function: f(x) = x² - 3.

  2. Calculate the Right-Hand Limit: We'll evaluate f(x) as x approaches 2 from the right (i.e., x ≥ 2). Here, we use the second part of the piecewise function: f(x) = (3x + 2) / 4.

  3. Compare the Limits: If the left-hand limit equals the right-hand limit, then the limit exists and is equal to that value. If the limits are not equal, then the limit does not exist.

Calculating the Left-Hand Limit

For the left-hand limit, we consider values of x that are slightly less than 2. Since x < 2, we use the first part of the function: f(x) = x² - 3. Now, let's plug in x = 2 into this part of the function: f(2) = 2² - 3 = 4 - 3 = 1. So, the left-hand limit is 1. This means as x gets closer and closer to 2 from the left side, the function's value approaches 1.

This is a crucial step because it helps us determine the value the function approaches from one direction. This value will tell us what value we should expect. The left-hand limit helps establish the behavior of the function. Remember, we're looking at the behavior of the function, not necessarily the value of the function at x = 2. The left-hand limit gives us a clear indication of what the function is 'aiming' for as we approach 2 from the left. This concept is a great step for this problem, which is why it’s important for us to have a clear understanding of what we are doing and the concept.

Finding the Right-Hand Limit

Now, let’s find the right-hand limit. We'll consider values of x slightly greater than or equal to 2. Because x ≥ 2, we use the second part of the function: f(x) = (3x + 2) / 4. Plug in x = 2: f(2) = (3 * 2 + 2) / 4 = (6 + 2) / 4 = 8 / 4 = 2. So, the right-hand limit is 2. This tells us that as x gets closer and closer to 2 from the right side, the function’s value approaches 2.

This step is very important in understanding the function's full behavior. This limit helps determine if the function converges to the same value from both directions. Calculating the right-hand limit is important for our understanding of how the function behaves. This understanding is crucial for solving the limit problem. This provides a complete picture of how the function behaves. This allows us to determine if the function is continuous or if there’s a jump discontinuity at x = 2. Without this information, we'd only have half the story, making it impossible to solve the problem.

Determining the Limit

Now, let's compare our left-hand and right-hand limits:

  • Left-hand limit: 1
  • Right-hand limit: 2

Since the left-hand limit (1) does not equal the right-hand limit (2), the limit of f(x) as x approaches 2 does not exist. This means the function f(x) doesn't have a single, defined value as x approaches 2. The function has a jump discontinuity at x = 2. The function approaches different values from the left and right sides, it has a jump discontinuity at x = 2. It’s like the function “jumps” from one value to another at that point.

Conclusion

Therefore, based on our calculations, the correct answer is that the limit does not exist. However, given the multiple-choice options, there seems to be an error in the options provided, as none of them correspond to our finding that the limit does not exist. If we were to select from the given options, the closest value to any of the one-sided limits would be option (B) 1, which is the left-hand limit. But, keep in mind, this is not technically the correct answer since the limit does not exist.

So, to recap, we've learned how to find limits for piecewise functions, especially by looking at both sides of the point where the function changes. We saw that sometimes limits don't exist, which is perfectly okay! Keep practicing, and you'll get the hang of it. Don't be afraid to ask questions and go back over the steps. That's how we all learn. Keep at it, and soon you will have a great understanding of the calculus limit concept, which is very important for the upcoming problems. Keep up the good work, and you'll be acing these problems in no time, guys!

Understanding these concepts is key to unlocking more advanced topics in calculus. So, stay curious and keep learning!