Evaluating Limit: \( \lim_{t \to 0^+} \frac{\sqrt{t}}{\cos(\sqrt{t})} \)

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Hey guys! Today, we're diving into a fascinating limit problem from the realm of calculus. We're tasked with evaluating the limit of the function tcos⁡(t){ \frac{\sqrt{t}}{\cos(\sqrt{t})} } as t{ t } approaches 0 from the positive side. This might seem a bit daunting at first, but don't worry, we'll break it down step by step. Understanding limits is crucial in calculus as they form the foundation for concepts like derivatives and integrals. Before we jump into the solution, let's understand why limits are so important. Limits help us describe the behavior of functions as they approach certain values, even if the function is not defined at that exact point. This is particularly useful when dealing with indeterminate forms or singularities. So, grab your thinking caps, and let's get started!

Understanding the Limit

When we talk about limits, we're essentially asking: what value does a function approach as its input gets closer and closer to a specific point? In this case, we want to know what happens to the function tcos⁡(t){ \frac{\sqrt{t}}{\cos(\sqrt{t})} } as t{ t } gets closer and closer to 0, but only from the positive side (that's what the 0+{ 0^+ } means). This is a one-sided limit, specifically a right-hand limit. To really grasp what's going on, let’s think about what happens to each part of the function as t{ t } approaches 0 from the positive side.

  • Numerator: The numerator is t{ \sqrt{t} }. As t{ t } gets closer to 0, t{ \sqrt{t} } also gets closer to 0. This is pretty straightforward. The square root of a very small positive number is also a very small positive number.
  • Denominator: The denominator is cos⁡(t){ \cos(\sqrt{t}) }. As t{ t } approaches 0, t{ \sqrt{t} } approaches 0 as well. So, we're looking at cos⁡(0){ \cos(0) }. Remember from trigonometry that cos⁡(0)=1{ \cos(0) = 1 }. This is a key piece of information.

So, we have the numerator approaching 0 and the denominator approaching 1. This gives us a good indication of what the overall limit will be. Let's dive into the formal evaluation to confirm our intuition.

Evaluating the Limit

Now, let's formally evaluate the limit. We have: lim⁡t→0+tcos⁡(t){ \lim_{t \to 0^+} \frac{\sqrt{t}}{\cos(\sqrt{t})} }

As we discussed, as t{ t } approaches 0 from the positive side:

  • t{ \sqrt{t} } approaches 0.
  • cos⁡(t){ \cos(\sqrt{t}) } approaches cos⁡(0)=1{ \cos(0) = 1 }.

We can use the limit properties that tell us the limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. In our case, the limit of the denominator is 1, so we're good to go. We can write: lim⁡t→0+tcos⁡(t)=lim⁡t→0+tlim⁡t→0+cos⁡(t)=01=0{ \lim_{t \to 0^+} \frac{\sqrt{t}}{\cos(\sqrt{t})} = \frac{\lim_{t \to 0^+} \sqrt{t}}{\lim_{t \to 0^+} \cos(\sqrt{t})} = \frac{0}{1} = 0 }

Therefore, the limit of the function as t{ t } approaches 0 from the positive side is 0. This confirms our initial intuition based on the behavior of the numerator and denominator. The crucial part here is recognizing how each component of the function behaves as t{ t } gets closer to 0.

Step-by-Step Solution

To make sure we've got this down pat, let's break down the solution into a step-by-step process. This will help solidify our understanding and make it easier to tackle similar problems in the future.

  1. Identify the Limit: We're given the limit lim⁡t→0+tcos⁡(t){ \lim_{t \to 0^+} \frac{\sqrt{t}}{\cos(\sqrt{t})} }.
  2. Analyze the Numerator: As t{ t } approaches 0 from the positive side, t{ \sqrt{t} } approaches 0.
  3. Analyze the Denominator: As t{ t } approaches 0 from the positive side, t{ \sqrt{t} } approaches 0, so cos⁡(t){ \cos(\sqrt{t}) } approaches cos⁡(0)=1{ \cos(0) = 1 }.
  4. Apply Limit Properties: Use the property that the limit of a quotient is the quotient of the limits (when the denominator's limit is not zero): lim⁡t→0+tcos⁡(t)=lim⁡t→0+tlim⁡t→0+cos⁡(t){ \lim_{t \to 0^+} \frac{\sqrt{t}}{\cos(\sqrt{t})} = \frac{\lim_{t \to 0^+} \sqrt{t}}{\lim_{t \to 0^+} \cos(\sqrt{t})} }
  5. Substitute the Limits: lim⁡t→0+tlim⁡t→0+cos⁡(t)=01{ \frac{\lim_{t \to 0^+} \sqrt{t}}{\lim_{t \to 0^+} \cos(\sqrt{t})} = \frac{0}{1} }
  6. Simplify: 01=0{ \frac{0}{1} = 0 }
  7. Conclusion: Therefore, lim⁡t→0+tcos⁡(t)=0{ \lim_{t \to 0^+} \frac{\sqrt{t}}{\cos(\sqrt{t})} = 0 }.

By following these steps, you can confidently evaluate similar limits. Remember, the key is to analyze the behavior of the function's components and apply the appropriate limit properties. And always double-check that the conditions for applying those properties are met!

Common Mistakes to Avoid

When dealing with limits, it's easy to make a few common mistakes. Being aware of these pitfalls can save you from headaches and help you get the correct answer every time. Let's take a look at some of these common errors:

  1. Ignoring the One-Sided Limit: In our problem, we're dealing with a right-hand limit (t→0+{ t \to 0^+ }). It's crucial to pay attention to the direction from which the variable is approaching the limit point. If we were considering a left-hand limit (t→0−{ t \to 0^- }), the square root function t{ \sqrt{t} } would not be defined for real numbers, making the limit undefined. Always check if the limit is one-sided and if it affects the function's definition.
  2. Assuming 00{ \frac{0}{0} } is Always 0: Ah, the infamous indeterminate form. While it's true that 01=0{ \frac{0}{1} = 0 }, the expression 00{ \frac{0}{0} } is indeterminate. This means it could be any number, or the limit might not exist at all. When you encounter 00{ \frac{0}{0} }, you need to use techniques like L'Hôpital's Rule, factoring, or rationalizing to simplify the expression before evaluating the limit. In our case, we didn't have 00{ \frac{0}{0} } because the denominator approached 1, not 0.
  3. Forgetting to Check the Denominator: We can only use the property lim⁡f(x)g(x)=lim⁡f(x)lim⁡g(x){ \lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)} } if the limit of the denominator, lim⁡g(x){ \lim g(x) }, is not equal to 0. If the denominator's limit is 0, we're back in indeterminate form territory, and we need a different approach. In our problem, the limit of the denominator was 1, so we were safe.
  4. Misunderstanding Trigonometric Limits: Trigonometric functions can sometimes trip us up. It's essential to remember key trigonometric limits, such as lim⁡x→0sin⁡(x)x=1{ \lim_{x \to 0} \frac{\sin(x)}{x} = 1 } and lim⁡x→0cos⁡(x)=1{ \lim_{x \to 0} \cos(x) = 1 }. In our case, we needed to know that cos⁡(0)=1{ \cos(0) = 1 }. A solid understanding of trig functions is crucial for calculus.
  5. Skipping Steps: It's tempting to jump to the answer, especially when a problem seems straightforward. However, skipping steps can lead to errors. Write out each step clearly, especially when you're learning. This will help you catch mistakes and reinforce your understanding.

By avoiding these common mistakes, you'll be well on your way to mastering limits! Remember, practice makes perfect, so keep solving problems and honing your skills.

Real-World Applications of Limits

Okay, so we've conquered this limit problem, but you might be wondering,