Solving A Limit: (1 - Cos(b/x)) / ((1/x) Tan(a/x)) = 8/3

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Hey guys! Today, we're diving deep into a fascinating limit problem that might seem daunting at first glance but becomes incredibly manageable once we break it down. We're going to explore how to evaluate the following limit:

lim (x→∞) [1 - cos(b/x)] / [(1/x) * tan(a/x)] = 8/3

This limit is a classic example that combines trigonometric functions, limits at infinity, and a touch of algebraic manipulation. So, buckle up, and let's get started!

Understanding the Problem

Before we jump into the solution, let's take a moment to understand what this problem is asking. We have a limit as x approaches infinity. This means we want to know what happens to the expression [1 - cos(b/x)] / [(1/x) * tan(a/x)] as x gets incredibly large. The constants a and b add a layer of generality, making the solution applicable to a range of similar problems.

Limits involving trigonometric functions often require us to use trigonometric identities and special limit rules. One of the key rules we'll be using is the small-angle approximation, which states that for small angles (in radians), sin(x) ≈ x and tan(x) ≈ x. This approximation will be crucial in simplifying our expression.

Our goal is to manipulate the given expression in such a way that we can apply these small-angle approximations and other limit rules to find the final value. The fact that the limit is equal to 8/3 gives us a target to aim for and will ultimately help us solve for the relationship between a and b.

Breaking Down the Solution

1. Initial Assessment and Simplification

The first thing we notice is that as x approaches infinity, both b/x and a/x approach zero. This is fantastic because it allows us to think about using small-angle approximations. However, directly substituting x = ∞ leads to an indeterminate form (0/0), which means we need to do some more work.

Let's start by rewriting the expression to make it a bit clearer:

lim (x→∞) [1 - cos(b/x)] / [(1/x) * tan(a/x)]

We can multiply both the numerator and denominator by x to get rid of the fraction in the denominator:

lim (x→∞) x * [1 - cos(b/x)] / tan(a/x)

2. Using Trigonometric Identities

The term 1 - cos(b/x) is a classic setup for using the double-angle formula. Recall the identity:

1 - cos(2θ) = 2sin²(θ)

We can rewrite 1 - cos(b/x) using this identity by letting 2θ = b/x, so θ = b/(2x):

1 - cos(b/x) = 2sin²(b/(2x))

Substituting this back into our limit expression, we get:

lim (x→∞) x * [2sin²(b/(2x))] / tan(a/x)

3. Applying Small-Angle Approximations

Now, we're in a great position to use the small-angle approximations. As x approaches infinity, b/(2x) and a/x approach zero. Therefore, we can approximate:

sin(b/(2x)) ≈ b/(2x)
tan(a/x) ≈ a/x

Substituting these approximations into our limit expression, we have:

lim (x→∞) x * [2 * (b/(2x))²] / (a/x)

4. Simplifying the Expression

Let's simplify the expression:

lim (x→∞) x * [2 * b² / (4x²)] / (a/x)

lim (x→∞) x * [b² / (2x²)] / (a/x)

lim (x→∞) [b² / (2x)] / (a/x)

Now, we can divide by a fraction by multiplying by its reciprocal:

lim (x→∞) [b² / (2x)] * (x/a)

lim (x→∞) b²x / (2ax)

The x terms cancel out:

lim (x→∞) b² / (2a)

5. Evaluating the Limit

Since there's no x left in the expression, the limit is simply:

b² / (2a)

We are given that this limit is equal to 8/3:

b² / (2a) = 8/3

6. Solving for the Relationship Between a and b

Now we have an equation relating a and b. Let's solve for the relationship. We can cross-multiply:

3b² = 16a

This gives us the relationship between a and b:

a = (3/16)b²

The Final Answer

So, we've successfully evaluated the limit and found the relationship between a and b. The key steps were using trigonometric identities, applying small-angle approximations, and simplifying the resulting expression. The final answer is that the limit equals 8/3 when a = (3/16)b².

Key Takeaways

  1. Small-Angle Approximations: These are incredibly powerful tools for evaluating limits involving trigonometric functions as the angle approaches zero. Remember that sin(x) ≈ x and tan(x) ≈ x for small x (in radians).
  2. Trigonometric Identities: Knowing your trigonometric identities is crucial. The identity 1 - cos(2θ) = 2sin²(θ) was key to simplifying the expression in this problem.
  3. Algebraic Manipulation: Don't be afraid to manipulate the expression. Multiplying by conjugates, using identities, and simplifying fractions are all essential techniques.
  4. Indeterminate Forms: Recognizing indeterminate forms (like 0/0) is the first step in knowing that you need to do more work to evaluate the limit.

Practice Problems

To solidify your understanding, try solving similar problems. Here are a few suggestions:

  1. Evaluate the limit: lim (x→0) [1 - cos(x)] / x²
  2. Evaluate the limit: lim (x→∞) x * sin(1/x)
  3. Find the relationship between a and b if lim (x→0) [tan(ax)] / [sin(bx)] = 2

Conclusion

Limits can seem intimidating, but by breaking them down into smaller steps and using the right tools, they become much more manageable. This problem showcased the power of trigonometric identities and small-angle approximations. Keep practicing, and you'll become a limit-solving pro in no time! Remember, the key is to understand the underlying concepts and apply them systematically. Good luck, guys, and happy problem-solving!