Limit Of (x*tan(x))/(1-cos(2x)) As X->0: Explained!

Oct 16, 2025 by ADMIN 52 views

Finding the Limit of (x*tan(x))/(1-cos(2x)) as x Approaches 0

Hey guys! Let's dive into a classic calculus problem today: finding the limit of a trigonometric function as x approaches 0. Specifically, we're tackling the limit of the function (xtan(x))/(1-cos(2x))* as x gets super close to zero. This type of problem often pops up in calculus courses, and mastering it is a key step in understanding limits and trigonometric identities. So, buckle up, and let's break it down together!

Understanding the Problem

Before we jump into the solution, let's make sure we really understand what the problem is asking. We're dealing with a limit, which is a fundamental concept in calculus. Think of a limit as the value a function approaches as its input gets closer and closer to a specific value. In this case, our input is x, and we want to see what happens to the function (xtan(x))/(1-cos(2x))* as x gets closer and closer to 0. It's like we're zooming in on the graph of the function near x = 0 to see where it's heading.

The function itself involves trigonometric functions – tan(x) and cos(2x). So, we'll likely need to remember some of our trig identities and how these functions behave near zero. A good first step in any limit problem is to try direct substitution. If we plug in x = 0 directly, we get (0 * tan(0))/(1 - cos(0)) = (0 * 0)/(1 - 1) = 0/0. Uh oh! This is an indeterminate form. It doesn't tell us anything about the actual limit. It just means we need to use some algebraic or trigonometric manipulation to simplify the expression before we can evaluate the limit. This is where the fun begins! We need to use our mathematical toolkit to rewrite the expression in a way that doesn't result in 0/0 when we substitute x = 0. Think of it like trying to open a lock – we need the right combination of techniques to unlock the limit.

Key Concepts and Tools

Okay, so we've identified the problem. Now, let's arm ourselves with the tools we need to solve it. Several key concepts and trigonometric identities will be crucial here:

Small Angle Approximations: For small values of x (close to 0), we have some really helpful approximations: sin(x) ≈ x and tan(x) ≈ x. These approximations come from the behavior of the sine and tangent functions near the origin, and they can be lifesavers in limit problems. They allow us to replace trigonometric functions with simpler algebraic expressions when we're dealing with limits as x approaches 0.
Trigonometric Identities: We'll definitely need to bust out some trig identities to simplify the expression. The most important one for this problem is the double-angle identity for cosine: cos(2x) = 1 - 2sin²(x). This identity is super useful for dealing with expressions involving cos(2x), as it allows us to rewrite it in terms of sin²(x). Remember, trigonometric identities are like the Swiss Army knives of trigonometry – they give us different ways to express the same thing, which can be incredibly handy when solving problems.
Limit Laws: We can use limit laws to break down the limit into smaller, more manageable pieces. For instance, the limit of a product is the product of the limits (provided the individual limits exist). These laws provide a framework for manipulating limits and making them easier to evaluate. Think of them as the rules of the game when you're working with limits.

With these tools in our arsenal, we're ready to tackle the problem head-on!

Solving the Limit

Alright, let's put our knowledge into action and solve this limit! Remember, our goal is to rewrite the expression so that we can evaluate the limit without getting an indeterminate form.

Apply the Double-Angle Identity: The first thing we'll do is use the double-angle identity for cosine to rewrite the denominator. We know that cos(2x) = 1 - 2sin²(x). Substituting this into our expression, we get:
```
lim (x→0) [x*tan(x) / (1 - cos(2x))] = lim (x→0) [x*tan(x) / (1 - (1 - 2sin²(x)))]
```
Simplifying the denominator, we have:
```
lim (x→0) [x*tan(x) / (2sin²(x))]
```
See how that identity helped us? We've gotten rid of the cos(2x) term and now have an expression involving sin²(x).
Rewrite tan(x) and Simplify: Next, let's rewrite tan(x) as sin(x)/cos(x). This will allow us to further simplify the expression:
```
lim (x→0) [x * (sin(x)/cos(x)) / (2sin²(x))]
```
Now we can simplify by canceling out one factor of sin(x) from the numerator and denominator:
```
lim (x→0) [x / (2sin(x)cos(x))]
```
We're making progress! The expression is looking cleaner already.
Use the Small Angle Approximation: Here's where our small angle approximation comes in handy. As x approaches 0, sin(x) ≈ x. Let's substitute this approximation into our expression:
```
lim (x→0) [x / (2 * x * cos(x))]
```
We can now cancel out the x terms:
```
lim (x→0) [1 / (2cos(x))]
```
Fantastic! The expression has simplified beautifully. We've eliminated the indeterminate form.
Evaluate the Limit: Finally, we can evaluate the limit by direct substitution. As x approaches 0, cos(x) approaches cos(0) = 1. So, we have:
```
lim (x→0) [1 / (2cos(x))] = 1 / (2 * 1) = 1/2
```
Therefore, the limit of (xtan(x))/(1-cos(2x))* as x approaches 0 is 1/2. Woohoo! We did it!

Summary and Key Takeaways

Let's recap what we've done and highlight the key takeaways from this problem:

We started with a limit problem that resulted in an indeterminate form (0/0) when we tried direct substitution. This meant we needed to use some clever techniques to simplify the expression.
We used the double-angle identity for cosine (cos(2x) = 1 - 2sin²(x)) to rewrite the denominator and get rid of the problematic cos(2x) term. This is a classic trick for dealing with expressions involving cos(2x) in limit problems.
We rewrote tan(x) as sin(x)/cos(x) to further simplify the expression and create opportunities for cancellation.
We applied the small angle approximation (sin(x) ≈ x for small x) to replace sin(x) with a simpler algebraic term. This approximation is a powerful tool for evaluating limits as x approaches 0.
By combining these techniques, we were able to simplify the expression to the point where we could directly substitute x = 0 and find the limit, which is 1/2.

This problem demonstrates the importance of having a strong understanding of trigonometric identities and limit laws. It also highlights the usefulness of small angle approximations in evaluating limits involving trigonometric functions. Remember, practice makes perfect! The more you work through these types of problems, the more comfortable you'll become with identifying the right techniques to use.

Practice Problems

Want to test your understanding? Try these similar problems:

Find the limit of (sin(x))/(x(1-cos(x))) as x approaches 0.
Find the limit of (x²)/(1-cos(x)) as x approaches 0.
Find the limit of (tan(x) - sin(x))/(x³) as x approaches 0.

These problems will give you a chance to apply the same techniques we used in this example. Good luck, and happy calculating!

Conclusion

So, there you have it! We've successfully navigated the limit of (xtan(x))/(1-cos(2x))* as x approaches 0. Remember, limits can seem tricky at first, but with the right tools and a bit of practice, you can conquer them! Keep exploring, keep learning, and most importantly, keep having fun with math! You've got this! This is a fundamental concept to grasp for more complex calculus problems, so make sure you feel confident with the steps we've outlined. Keep practicing, and you'll be a limit-solving pro in no time!