Finding Limits Of A Trigonometric Function

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Finding Limits of a Trigonometric Function

Hey guys! Today, we're diving into the world of limits and trigonometric functions. We'll be looking at how to evaluate the limits of a function as x approaches positive and negative infinity. Specifically, we'll be working with the function f(x) = 12 * arctan(5x + 4x³). This might seem a little intimidating at first, but trust me, with a few key concepts and a step-by-step approach, we'll crack this problem together. Let's get started, shall we?

Understanding the Basics: Limits and Inverse Tangent

Alright, before we jump into the calculation, let's quickly recap what limits and the inverse tangent function are all about. In simple terms, a limit describes the value a function approaches as the input (in this case, x) gets closer and closer to a certain value (in our case, infinity or negative infinity). It's like asking: "Where is this function heading?" even if it never actually gets there. This is super helpful because it tells us about the function's behavior near certain points, even if we can't directly plug those points into the function (like in the case of infinity).

Now, let's talk about the inverse tangent function, also known as arctan or tan⁻¹. This function is the opposite of the tangent function. While the tangent function takes an angle and returns a ratio, the arctan function takes a ratio and returns an angle. The crucial thing to remember for our problem is the range of the arctan function. The arctan function's output (the angle) always falls within the range of ( -π/2, π/2 ). This is super important! It means the values arctan spits out are always going to be bounded, no matter what we feed into it.

Now we've got the foundation laid, let's get into the main part of the problem. We want to find out what happens to f(x) as x goes towards both positive and negative infinity. This is where those basic concepts we just covered come into play! Keep in mind we are working with limits so the main idea is to observe the tendency of the function as the variable approaches a certain value.

The Inverse Tangent Function

Consider the arctan function; it has a range of (-π/2, π/2). The output of the arctan function is an angle. The inverse tangent function, or arctan, has a limited output range between -π/2 and π/2. This is super important because it constrains the function's potential values. No matter what input we give it, the arctan will always give us a value that remains within that range. Therefore, we can utilize this property to determine the limit of the given function. Let's delve in deeper!

Evaluating the Limit as x Approaches Positive Infinity

Okay, let's tackle the first part: finding lim (x→∞) f(x). This means, what happens to our function f(x) as x gets infinitely large in the positive direction? Remember, f(x) = 12 * arctan(5x + 4x³). Now, as x gets super huge (approaches infinity), the term 4x³ will dominate the expression 5x + 4x³. Why? Because the x³ term grows much, much faster than the x term. So, for very large values of x, 5x + 4x³ is essentially the same as 4x³.

This means that inside the arctan function, we essentially have something that's growing towards positive infinity. Think about it: a very large positive number (x) cubed (x³) is still a very large positive number. Multiply that by 4, and it becomes even bigger! So, 5x + 4x³ approaches positive infinity as x approaches positive infinity. Because of this, we know that arctan(5x + 4x³) is going to tend towards π/2. (Remember the range of the arctan function?).

Since, the arctan function approaches π/2 as x goes to infinity, we can replace that term with π/2, then our function becomes: f(x) = 12 * arctan(5x + 4x³) ≈ 12 * (π/2) = 6π.

So, as x approaches positive infinity, f(x) approaches 6π. Therefore, lim (x→∞) f(x) = 6π.

Step-by-step breakdown:

  1. Analyze the inner function: As x → ∞, the term 4x³ dominates 5x, so 5x + 4x³ → ∞.
  2. Evaluate the arctan: Since arctan(u) approaches π/2 as u → ∞.
  3. Apply the constant: Multiply the result by 12: 12 * (π/2) = 6π.
  4. Final answer: The limit is 6π.

Evaluating the Limit as x Approaches Negative Infinity

Alright, now for the second part: finding lim (x→-∞) f(x). This is similar to what we did before, but now we're looking at what happens as x gets infinitely large in the negative direction. Again, f(x) = 12 * arctan(5x + 4x³).

As x approaches negative infinity, the 4x³ term still dominates. But now, since we're cubing a large negative number, the result will also be a large negative number. This means that 5x + 4x³ approaches negative infinity as x approaches negative infinity.

So, inside the arctan function, we have something going towards negative infinity. Now we know that when the argument of the arctan function approaches negative infinity, the arctan function approaches -π/2 (remember the range?).

We also know that arctan(5x + 4x³) approaches -π/2, and then we just need to multiply by 12: f(x) = 12 * arctan(5x + 4x³) ≈ 12 * (-π/2) = -6π.

Therefore, as x approaches negative infinity, f(x) approaches -6π. So, lim (x→-∞) f(x) = -6π.

Step-by-step breakdown:

  1. Analyze the inner function: As x → -∞, 4x³ dominates 5x, so 5x + 4x³ → -∞.
  2. Evaluate the arctan: arctan(u) approaches -π/2 as u → -∞.
  3. Apply the constant: Multiply the result by 12: 12 * (-π/2) = -6π.
  4. Final answer: The limit is -6π.

Summary and Conclusion

So, to recap, guys! We've successfully evaluated the limits of our function f(x) = 12 * arctan(5x + 4x³). We found that:

  • lim (x→∞) f(x) = 6π
  • lim (x→-∞) f(x) = -6π

We did this by understanding how the arctan function behaves as its input approaches positive and negative infinity, and how the dominant terms in the expression within the arctan function affect the overall limit. The key here was recognizing that the range of the arctan function is critical in determining the limits. It's really that simple.

I hope this explanation was helpful! Calculus can seem tricky at first, but with a good grasp of the fundamentals and a step-by-step approach, you can conquer any limit problem! Keep practicing, and you'll be acing these problems in no time. If you have any more questions or want to explore other examples, just ask! Keep learning, keep exploring, and keep having fun with math! Bye for now, see ya!