Unlocking Limits: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the fascinating world of limits. Specifically, we'll tackle the problem of evaluating the limit: ${ \lim_{x \to 8} \frac{\sqrt{9+2x} - 5}{\sqrt[3]{x} - 2} }$. Don't worry, guys, it might look a little intimidating at first glance, but with the right approach, it's totally manageable. We're going to break it down step by step, making sure you grasp every detail. The main goal here is to make sure you fully understand not just the solution, but the reasoning behind it. So, let's get started and unravel this mathematical puzzle together. This exploration will not only help you solve this specific limit problem but will also equip you with valuable techniques applicable to a wide range of limit evaluations. Understanding limits is crucial in calculus, forming the foundation for concepts like derivatives and integrals.

Before we begin, remember that limits describe the behavior of a function as the input (in this case, x) approaches a certain value. In our problem, we're examining what happens to the function as x gets closer and closer to 8. Direct substitution is often the first thing we try. However, if we substitute x = 8 directly into the expression, we get an indeterminate form (0/0), which doesn't tell us much. This is where our clever techniques come into play! We need to manipulate the expression algebraically to eliminate this indeterminacy. This usually involves techniques like rationalizing the numerator or denominator, or using other algebraic tricks. The goal is to transform the expression into a form where we can directly substitute the value of x and obtain a meaningful answer. Remember, the beauty of mathematics lies in its logical structure, so each step we take must be based on solid mathematical principles. So, let's gear up and dive in, ready to conquer this limit problem!

Rationalizing the Numerator and Denominator: The Key Strategy

Alright, folks, our primary tactic here will be to rationalize both the numerator and the denominator. What does this mean? Basically, we want to get rid of the radicals (square roots and cube roots) in a strategic manner. This will involve some clever algebraic manipulation. Let's start with the numerator, which has a square root. To rationalize this, we'll multiply both the numerator and the denominator of the entire expression by the conjugate of the numerator. The conjugate of √9+2x - 5 is √9+2x + 5. Doing this will help us eliminate the square root in the numerator, utilizing the difference of squares identity: (a - b)(a + b) = a² - b². We want to multiply the top and bottom by √9+2x + 5 like so: ${ \frac{\sqrt{9+2x} - 5}{\sqrt[3]{x} - 2} \cdot \frac{\sqrt{9+2x} + 5}{\sqrt{9+2x} + 5} }$.

This gives us: ${ \frac{(9+2x) - 25}{(\sqrt[3]{x} - 2)(\sqrt{9+2x} + 5)} }$.

Simplify the numerator to get: ${ \frac{2x - 16}{(\sqrt[3]{x} - 2)(\sqrt{9+2x} + 5)} }$.

Now, let's focus on the denominator. We see a cube root, and to deal with that, we're going to use a similar, yet slightly different approach. This time, we'll employ the identity a³ - b³ = (a - b)(a² + ab + b²). The trick is to identify the 'a' and 'b' and then manipulate our expression. Notice that we want to get rid of the √[3]{x} - 2 term. We can rewrite the denominator. To do this, let's think of a as ∛x and b as 2. This means a³ = x and b³ = 8. Thus, we can say that the denominator should be multiplied by (∛x² + 2∛x + 4) / (∛x² + 2∛x + 4).

So, before that, we have ${ \frac{2x - 16}{(\sqrt[3]{x} - 2)(\sqrt{9+2x} + 5)} }$.

Now multiply the denominator by ∛x² + 2∛x + 4

This gives us: ${ \frac{2x - 16}{(\sqrt[3]{x} - 2)(\sqrt{9+2x} + 5)} \cdot \frac{\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4}{\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4} }$.

We then have: ${ \frac{2(x - 8)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4)}{((x - 8)(\sqrt{9+2x} + 5)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4))} }$.

Notice that the x - 8 term appears in both the numerator and the denominator, allowing us to cancel them out, which is exactly what we wanted to do in order to get rid of the zero over zero case. This simplification step is the essence of solving limit problems. Always strive to simplify the expression to a point where direct substitution becomes feasible. Remember, patience and persistence are key!

Simplifying and Evaluating: The Grand Finale

Great job, guys! After all that hard work, we're now at the final stretch. After the rationalization and simplification steps, we've got an expression that's much more manageable. Let's recap where we are after canceling out the (x-8) terms. We have:

${ \frac{2}{(\sqrt{9+2x} + 5)(\sqrt[3]{x^2} + 2\sqrt[3]{x} + 4)} }$.

Now, the moment of truth: we're going to substitute x = 8 into our simplified expression. This is where we see the fruits of our labor, as we should now be able to get a defined value! Substituting x = 8, we get:

${ \frac{2}{(\sqrt{9+2(8)} + 5)(\sqrt[3]{8^2} + 2\sqrt[3]{8} + 4)} }$.

Let's break that down further. First, we have √9+2(8) = √25 = 5. Then we've got: ∛8² + 2∛8 + 4 = ∛64 + 2(2) + 4 = 4 + 4 + 4 = 12. Putting it all together:

${ \frac{2}{(5 + 5)(4 + 4 + 4)} = \frac{2}{10 \cdot 12} = \frac{2}{120} = \frac{1}{60} }$.

Ta-da! We've found the limit! So, as x approaches 8, the function approaches 1/60. This result tells us about the behavior of the function near the point x = 8, even if the function might not be defined exactly at x = 8. It's a fundamental concept in calculus and lays the groundwork for understanding continuity and derivatives. The whole process underscores the importance of algebraic manipulation, understanding of mathematical identities, and, of course, careful calculation. Remember, practice makes perfect. The more limit problems you tackle, the more comfortable and confident you'll become. So, keep exploring, keep practicing, and keep that mathematical spirit alive! You've officially conquered this limit problem. Give yourself a pat on the back. It's time to celebrate with some more practice problems. Keep learning and growing!