Limit Evaluation: Y = (x+1)/(x^2-1) As X→1

Hey guys! Let's dive into the fascinating world of limits with a classic example. We're going to explore the function y = (x+1)/(x^2-1) and figure out what happens as x gets super close to 1. We’ll also fill out a table to get a visual sense of what's going on. So, grab your thinking caps, and let's get started!

Understanding Limits

Before we jump into the specifics, let's quickly recap what a limit actually is. In simple terms, a limit tells us what value a function approaches as its input (in this case, x) gets closer and closer to a specific value. It's not necessarily the value of the function at that point, but rather where it's heading. This is especially useful when dealing with functions that might be undefined at a particular point, like our function here when x is 1 (we’ll see why soon!).

The concept of limits is foundational in calculus, underpinning ideas like continuity, derivatives, and integrals. Mastering limits opens the door to understanding more complex mathematical concepts, making it a crucial skill for students and professionals in fields like engineering, physics, and computer science. The idea allows us to analyze the behavior of functions near points of interest, even where the function itself might not be well-defined. This is essential for solving problems involving rates of change, optimization, and approximations.

For instance, in physics, limits are used to define instantaneous velocity and acceleration. In engineering, they help in designing stable structures and systems. In computer science, limits are relevant in analyzing the efficiency of algorithms. Therefore, a solid grasp of limits is not just an academic exercise but a practical tool for problem-solving across various disciplines. Think of limits as the mathematical equivalent of zooming in on a graph. We're not concerned with the big picture, just the tiny area around a specific point. This zoomed-in view allows us to understand the function's behavior with great precision, revealing patterns and trends that might be obscured at a larger scale.

The Function: y = (x+1)/(x^2-1)

Our function for today is y = (x+1)/(x^2-1). Now, the first thing you might notice is that the denominator, x^2-1, becomes zero when x = 1. This means the function is undefined at x = 1. But don't worry, this is exactly why limits are so powerful! We can investigate what happens as x gets close to 1, without actually plugging in 1 itself.

Before we start plugging in values, let's see if we can simplify our function. This is a crucial step in evaluating limits, as simplification often reveals hidden behaviors and removes troublesome singularities. The denominator, x^2 - 1, is a difference of squares, which we can factor as (x + 1)(x - 1). So, our function becomes:

y = (x + 1) / ((x + 1)(x - 1)).

Now, we can cancel out the (x + 1) terms in the numerator and denominator, but with a tiny caveat. We can only do this if x ≠ -1, because dividing by zero is a big no-no. However, since we're interested in the limit as x approaches 1, and not -1, this simplification is perfectly valid for our purpose. After canceling, we get:

y = 1 / (x - 1), for x ≠ -1.

This simplified form is much easier to work with. It clearly shows that as x gets closer to 1, the denominator (x - 1) gets closer to 0. This suggests that the value of y will become very large, either positive or negative, depending on whether x approaches 1 from the left (values less than 1) or the right (values greater than 1).

Completing the Table

Now, let’s fill out the table to see this in action. We'll plug in the given x values into our simplified function, y = 1 / (x - 1).

Here's the table we need to complete:

Let's calculate the y values for each x:

• x = 0: y = 1 / (0 - 1) = -1
• x = 0.5: y = 1 / (0.5 - 1) = -2
• x = 0.9: y = 1 / (0.9 - 1) = -10
• x = 0.99: y = 1 / (0.99 - 1) = -100
• x = 1: We already know the function is undefined here, so we leave it blank.
• x = 1.01: y = 1 / (1.01 - 1) = 100
• x = 1.1: y = 1 / (1.1 - 1) = 10
• x = 1.5: y = 1 / (1.5 - 1) = 2
• x = 2: y = 1 / (2 - 1) = 1

Here’s the completed table:

Evaluating the Limit

Now, let’s analyze what the table tells us about the limit as x approaches 1. Notice what happens to the y values as x gets closer to 1 from the left (values less than 1) and from the right (values greater than 1):

• As x approaches 1 from the left (e.g., 0.9, 0.99): The y values become increasingly large and negative (-10, -100). This suggests the function is heading towards negative infinity.
• As x approaches 1 from the right (e.g., 1.01, 1.1): The y values become increasingly large and positive (100, 10). This suggests the function is heading towards positive infinity.

Since the function approaches different values (negative infinity from the left and positive infinity from the right), we say that the limit does not exist. For a limit to exist at a point, the function must approach the same value from both sides.

We can write this mathematically as:

• lim (x→1-) (x+1)/(x^2-1) = -∞ (Limit from the left)
• lim (x→1+) (x+1)/(x^2-1) = +∞ (Limit from the right)

Because the left-hand limit and the right-hand limit are not equal, the overall limit as x approaches 1 does not exist.

Graphical Interpretation

To further solidify our understanding, let's think about what the graph of y = 1 / (x - 1) looks like. It's a hyperbola with a vertical asymptote at x = 1. This means the graph gets infinitely close to the vertical line x = 1 but never actually touches it. As we approach x = 1 from the left, the graph plunges downwards towards negative infinity. As we approach x = 1 from the right, the graph shoots upwards towards positive infinity. This visual representation perfectly matches our numerical findings from the table and confirms that the limit does not exist at x = 1.

Graphical representations are invaluable in calculus for understanding the behavior of functions. They provide a visual check on analytical results, helping to identify asymptotes, discontinuities, and other critical features. In the case of limits, graphs vividly illustrate how a function behaves in the vicinity of a particular point, making it easier to grasp the concept of approaching a value without necessarily reaching it. For instance, the graph of y = 1 / (x - 1) clearly shows the function diverging to infinity on either side of x = 1, reinforcing our conclusion that the limit does not exist at this point. The steepness of the curve near the asymptote also provides insight into the rate at which the function increases or decreases, adding another layer of understanding beyond the numerical data in the table.

Conclusion

So, guys, we've successfully evaluated the limit of the function y = (x+1)/(x^2-1) as x approaches 1. By simplifying the function, creating a table of values, and analyzing the behavior from both sides, we've determined that the limit does not exist. This is because the function approaches different infinities depending on the direction from which x approaches 1.

Remember, understanding limits is a crucial step in mastering calculus. It allows us to analyze the behavior of functions at points where they might be undefined, and it forms the foundation for many other important concepts. Keep practicing, and you'll become a limit-evaluating pro in no time!

Key Takeaways:

• Limits describe the value a function approaches as its input approaches a specific value.
• Simplifying functions can make limit evaluation easier.
• A limit exists only if the function approaches the same value from both the left and the right.
• Tables and graphs are helpful tools for visualizing and understanding limits.

Now, go forth and conquer more limits! You got this!