Limit Calculation: (2x^2 Sin 3x Cos 2x) / (6x^3 + Tan^2 3x Sin 4x)

Hey guys! Ever stumbled upon a limit problem that looks like a monster? Today, we're going to dissect one such beast from the realm of trigonometry and calculus. We'll be tackling the limit of a function as x approaches 0, which involves sines, cosines, tangents, and powers of x. Buckle up, because it's going to be a mathematical rollercoaster! We'll break down the problem step-by-step, making sure everyone, from calculus newbies to seasoned math enthusiasts, can follow along. So, let's dive deep into the heart of this limit and emerge victorious with a solution in hand.

Understanding the Problem

The problem we're tackling is this: How do we calculate the limit of the function $\frac{2x^2\sin 3x\cos 2x}{6x^3+\tan^2 3x\sin 4x}$ as x approaches 0? At first glance, it looks intimidating with its mix of trigonometric functions and algebraic terms. However, don't let the complexity scare you! We can solve this by employing some clever techniques and trigonometric identities. The key is to remember that limits describe the behavior of a function as it gets close to a certain value, not necessarily the value at that point. This is crucial because plugging in x = 0 directly leads to an indeterminate form (0/0), which doesn't tell us anything about the limit. Instead, we need to manipulate the expression to reveal its true behavior near zero. This might involve using small-angle approximations, trigonometric identities, or other algebraic tricks. The goal is to transform the expression into a form where we can easily evaluate the limit. Remember, practice makes perfect, and with a bit of patience, even the most daunting limit problems can be conquered!

Breaking Down the Function

Let's break down the function piece by piece. We have $2x^2\sin 3x\cos 2x$ in the numerator and $6x^3+\tan^2 3x\sin 4x$ in the denominator. Each of these terms behaves differently as x approaches 0. Understanding how these individual components behave is crucial to figuring out the overall limit. For instance, we know that sin(x) behaves like x for small values of x, and similarly, tan(x) also behaves like x. These approximations will be vital later on. Also, notice the powers of x: we have $x^2$ in the numerator and $x^3$ in the denominator, which suggests that we might be able to simplify the expression by canceling out some terms. The cosine term, cos(2x), is a bit different; as x approaches 0, cos(2x) approaches 1. This is because the cosine function is continuous, and cos(0) = 1. Keeping track of these behaviors is like assembling the pieces of a puzzle; each piece contributes to the final solution. By carefully analyzing each term, we can develop a strategy for simplifying the entire expression and finding the limit.

Key Concepts and Theorems

Before we dive into the solution, let's refresh some key concepts and theorems. The most important concept here is the limit itself – understanding what it means for a function to approach a certain value. Limits are the foundation of calculus, so a solid grasp of this concept is essential. We also need to be familiar with trigonometric limits, especially the small-angle approximations: $\lim_{x\to 0} \frac{\sin x}{x} = 1$ and $\lim_{x\to 0} \frac{\tan x}{x} = 1$. These are our secret weapons for dealing with trigonometric functions as x approaches 0. Another crucial theorem is the limit of a product and quotient: the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits (provided the limits exist and the denominator's limit is not zero). Finally, we'll use the fact that the cosine function is continuous, meaning that $\lim_{x\to a} \cos x = \cos a$. With these tools in our mathematical arsenal, we're well-equipped to tackle the problem at hand.

Small-Angle Approximations

The small-angle approximations are the stars of the show in this problem. As x approaches 0, sin(x) gets incredibly close to x, and tan(x) does too. Mathematically, we write this as $\sin x \approx x$ and $\tan x \approx x$ for small x. But why is this so important? These approximations allow us to replace trigonometric functions with simpler algebraic expressions, making the limit much easier to evaluate. Think of it like swapping out a complex gear in a machine for a simpler one, making the whole mechanism run smoother. For our specific problem, this means we can replace sin(3x) with 3x, tan(3x) with 3x, and sin(4x) with 4x when x is close to 0. This dramatically simplifies the expression and allows us to cancel out terms and ultimately find the limit. Remember, these approximations are valid only when x is close to 0, so it's crucial to ensure this condition is met before applying them.

Solving the Limit

Now, let's get our hands dirty and solve the limit! We start with the original expression: $\lim_{x\to 0}\frac{2x^2\sin 3x\cos 2x}{6x^3+\tan^2 3x\sin 4x}$. Our first move is to apply the small-angle approximations. Replacing sin(3x) with 3x, tan(3x) with 3x, and sin(4x) with 4x, we get: $\lim_{x\to 0}\frac{2x^2(3x)\cos 2x}{6x^3+(3x)^2(4x)}$. Notice how the trigonometric functions have transformed into simple algebraic terms. This is the power of small-angle approximations! Next, we simplify the expression: $\lim_{x\to 0}\frac{6x^3\cos 2x}{6x^3+36x^3}$. We can factor out $x^3$ from both the numerator and the denominator: $\lim_{x\to 0}\frac{6x^3\cos 2x}{x^3(6+36)}$. Now, we can cancel out the $x^3$ terms: $\lim_{x\to 0}\frac{6\cos 2x}{6+36}$. This leaves us with a much simpler limit. As x approaches 0, cos(2x) approaches cos(0), which is 1. So, the limit becomes $\frac{6(1)}{42}$, which simplifies to $\frac{1}{7}$. Voila! We've successfully calculated the limit.

Step-by-Step Calculation

Let's walk through the calculation step-by-step to make sure everything is crystal clear.

1. Original Limit: $\lim_{x\to 0}\frac{2x^2\sin 3x\cos 2x}{6x^3+\tan^2 3x\sin 4x}$
2. Apply Small-Angle Approximations: $\lim_{x\to 0}\frac{2x^2(3x)\cos 2x}{6x^3+(3x)^2(4x)}$
3. Simplify: $\lim_{x\to 0}\frac{6x^3\cos 2x}{6x^3+36x^3}$
4. Factor out $x^3$: $\lim_{x\to 0}\frac{6x^3\cos 2x}{x^3(6+36)}$
5. Cancel $x^3$: $\lim_{x\to 0}\frac{6\cos 2x}{6+36}$
6. Evaluate $\cos 2x$ as $x \to 0$: $\frac{6\cos(0)}{42}$
7. Simplify: $\frac{6(1)}{42} = \frac{1}{7}$

Each step builds upon the previous one, transforming the complex original limit into a straightforward calculation. Notice how the small-angle approximations are the key to unlocking the solution. They allow us to bypass the indeterminate form and reveal the true behavior of the function near x = 0.

Common Mistakes to Avoid

When tackling limit problems, it's easy to stumble into common pitfalls. Let's highlight some mistakes to avoid. One frequent error is applying small-angle approximations incorrectly. Remember, these approximations are valid only when x is close to 0. Don't use them for large values of x! Another mistake is forgetting to check for indeterminate forms. If you plug in the limit value and get 0/0 or ∞/∞, it doesn't mean the limit doesn't exist; it just means you need to do more work. Algebraic errors are also common, especially when simplifying complex expressions. Double-check your algebra to avoid mistakes. Finally, be careful with trigonometric identities. Make sure you're using the correct identities and applying them appropriately. By being aware of these common errors, you can significantly improve your accuracy and confidence in solving limit problems. Math it's not about being perfect the first time, but learning from errors!

Importance of Showing Your Work

Showing your work isn't just about pleasing your math teacher; it's a crucial step in the problem-solving process. When you write down each step, you're forced to think through your reasoning and identify potential errors. It's like creating a roadmap for your solution, making it easier to navigate the problem. Showing your work also helps you understand the underlying concepts better. Each step is a building block, and by writing them down, you reinforce your understanding of how the pieces fit together. Moreover, if you do make a mistake, showing your work makes it easier to pinpoint the error. It's like leaving a trail of breadcrumbs that leads you back to the source of the problem. In the context of exams or assignments, showing your work can also earn you partial credit, even if you don't arrive at the final answer. So, always show your work – it's a win-win situation!

Conclusion

So, there you have it! We've successfully navigated a tricky limit problem using small-angle approximations and a bit of algebraic manipulation. Remember, the key to mastering limits (and math in general) is practice and understanding the underlying concepts. Don't be afraid to break down complex problems into smaller, manageable steps. And most importantly, don't give up! Every problem you solve is a step forward in your mathematical journey. Keep practicing, keep exploring, and you'll become a limit-solving pro in no time. You've got this, guys! Now go out there and conquer those mathematical mountains! This journey through the intricacies of limit calculations highlights not only the mechanical steps involved but also the importance of understanding the underlying principles. From recognizing the applicability of small-angle approximations to meticulously simplifying expressions, each step is a testament to the power of methodical problem-solving. It’s not just about arriving at the correct answer but also about appreciating the elegance and logic of mathematics. Embracing challenges like this fosters a deeper understanding and a greater appreciation for the beauty inherent in mathematical concepts.