Calculate The Limit: A Step-by-Step Guide

Hey guys! Let's dive into calculating limits, specifically looking at the expression (x² + x + 2) / (x - 3 - x² - 2x - 3). This might look a bit scary at first, but don't worry, we'll break it down into manageable steps. Understanding limits is super important in calculus, and it's a foundational concept for more advanced topics. So, buckle up, and let's get started!

Understanding the Basics of Limits

Before we jump into the specific calculation, let's quickly recap what limits are all about. 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 particular value. It's not necessarily the value of the function at that point, but rather where the function is heading. Think of it like this: you're walking towards a door, the limit is the door itself, even if you never quite reach it. The limit exists if, as x approaches a certain value from both the left and the right, the function approaches the same value. If the function approaches different values from different directions, or if it goes to infinity, the limit does not exist.

Limits are crucial because they help us deal with situations where direct substitution doesn't work. For example, if we tried to directly substitute a value that makes the denominator of our expression zero, we'd end up with an undefined result. That's where the magic of limits comes in! We use various techniques like factoring, rationalizing, and L'Hôpital's Rule (which we'll touch on later) to manipulate the expression and find the limit. Mastering these techniques is essential for tackling more complex calculus problems. Remember, the goal is to find out what the function approaches, not necessarily what it equals at a specific point.

Simplifying the Expression

Okay, now let's get our hands dirty with the given expression: (x² + x + 2) / (x - 3 - x² - 2x - 3). The first thing we should always do is simplify it as much as possible. This makes the subsequent steps much easier. Combining like terms in the denominator gives us:

(x² + x + 2) / (-x² - x - 6)

Notice that both the numerator and the denominator are quadratic expressions. While we could try to factor them, in this case, they don't factor nicely with real numbers. This means we'll likely need to use other techniques to evaluate the limit.

Determining the Approach Value

Here's a crucial piece of information missing from the original question: what value of 'x' are we approaching? The limit is always calculated as x approaches a specific number (or infinity). Without this, we can't actually calculate a numerical limit. Let's consider a few different scenarios to illustrate the process:

Scenario 1: x approaches a finite number (e.g., x approaches 2)

If we were asked to find the limit as x approaches 2, we would first try direct substitution:

((2)² + 2 + 2) / (-(2)² - 2 - 6) = (4 + 2 + 2) / (-4 - 2 - 6) = 8 / -12 = -2/3

In this case, direct substitution works perfectly fine! The function is defined at x = 2, and the limit is simply the value of the function at that point.

Scenario 2: x approaches a value that makes the denominator zero

This is where things get more interesting. Let's find the values of x that make the denominator -x² - x - 6 equal to zero. We can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = -1, b = -1, and c = -6.

x = (1 ± √((-1)² - 4(-1)(-6))) / (2 * -1)

x = (1 ± √(1 - 24)) / -2

x = (1 ± √(-23)) / -2

Since the discriminant (the value inside the square root) is negative, there are no real values of x that make the denominator zero. This means we don't have to worry about division by zero for any real number that x approaches.

Scenario 3: x approaches infinity (or negative infinity)

Now, let's consider what happens as x becomes very, very large (positive or negative). This is where we look at the highest powers of x in the numerator and denominator. In our expression, the highest power is x².

(x² + x + 2) / (-x² - x - 6)

To evaluate the limit as x approaches infinity, we can divide both the numerator and denominator by x²:

(1 + 1/x + 2/x²) / (-1 - 1/x - 6/x²)

As x approaches infinity, 1/x and 2/x² both approach zero. Therefore, the expression simplifies to:

(1 + 0 + 0) / (-1 - 0 - 0) = 1 / -1 = -1

So, the limit as x approaches infinity (or negative infinity) is -1.

Using L'Hôpital's Rule (If Applicable)

L'Hôpital's Rule is a powerful tool for evaluating limits of the form 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches c is of the form 0/0 or ∞/∞, then:

lim (x→c) f(x)/g(x) = lim (x→c) f'(x)/g'(x)

where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.

In our case, the expression is (x² + x + 2) / (-x² - x - 6). As we saw earlier, direct substitution only results in an indeterminate form (like 0/0 or ∞/∞) when x approaches infinity. We already solved for the infinity case above. However, to illustrate L'Hopital's rule, let's apply it to the limit as x approaches infinity:

f(x) = x² + x + 2, f'(x) = 2x + 1 g(x) = -x² - x - 6, g'(x) = -2x - 1

lim (x→∞) (2x + 1) / (-2x - 1)

We still have ∞/∞ form. So let's apply L'Hopital's rule again:

f''(x) = 2 g''(x) = -2

lim (x→∞) 2 / -2 = -1

We get the same answer, -1. L'Hôpital's Rule can be very helpful, but remember to check if the conditions for applying it are met first!

Summary and Key Takeaways

Alright, guys, let's recap what we've learned about calculating limits:

1. Simplify the expression: Combine like terms and look for opportunities to factor.
2. Determine the approach value: What value is x approaching? This is crucial!
3. Try direct substitution: If it works, you're done!
4. Check for division by zero: If the denominator becomes zero, you'll need to use other techniques.
5. Consider limits at infinity: Divide by the highest power of x and see what happens as x gets very large.
6. L'Hôpital's Rule: If you have an indeterminate form (0/0 or ∞/∞), this can be a powerful tool.

Remember, practice makes perfect! The more you work with limits, the more comfortable you'll become with these techniques. Don't be afraid to make mistakes – that's how we learn! And always double-check your work to avoid careless errors.

Limits are a fundamental concept in calculus, and understanding them is essential for success in higher-level math courses. So, keep practicing, and you'll be a limit-calculating pro in no time! Good luck, and happy calculating!