L'Hôpital's Rule: Evaluating Limits Example

Hey guys! Today, we're diving into a classic calculus problem: evaluating limits using L'Hôpital's Rule. This rule is super handy when we encounter indeterminate forms like 0/0 or ∞/∞. We'll break down a specific example step by step, so you can see exactly how it works. Let's get started!

Understanding the Limit Problem

Before we jump into L'Hôpital's Rule, let's take a look at the limit we're trying to solve. Our goal is to find:

$\lim _{x \rightarrow 0} \frac{4 \cos (7 x)-4}{11 x}$

When we directly substitute x = 0 into the expression, we get:

$\frac{4 \cos (7 * 0)-4}{11 * 0} = \frac{4 \cos (0)-4}{0} = \frac{4 * 1 - 4}{0} = \frac{0}{0}$

Ah-ha! We've got an indeterminate form of 0/0. This is our cue that L'Hôpital's Rule might be the perfect tool for the job. 0/0 is one of the forms where L'Hôpital’s Rule shines, signaling that we can differentiate the numerator and the denominator separately and then re-evaluate the limit. So, let’s explore what L'Hôpital's Rule is all about and how we can apply it effectively to crack this limit problem.

What is L'Hôpital's Rule?

Alright, so what exactly is L'Hôpital's Rule? In simple terms, it's a rule that helps us evaluate limits of functions that result in indeterminate forms. These forms usually pop up when we try to directly substitute a value into a limit and end up with something like 0/0 or ∞/∞. Basically, L'Hôpital's Rule states that if we have a limit of the form:

$\lim_{x \to c} \frac{f(x)}{g(x)}$

and both f(x) and g(x) approach 0 or both approach ∞ as x approaches c, then:

$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

...provided the limit on the right-hand side exists. What this means is that instead of directly evaluating the original limit, we can take the derivative of the numerator (f(x)) and the derivative of the denominator (g(x)) separately, and then evaluate the limit of the new fraction. This often simplifies the expression and makes the limit easier to find. The heart of L'Hôpital's Rule lies in this elegant transformation, allowing us to tackle seemingly complex limits by differentiating the numerator and the denominator. It's like having a secret weapon in our calculus toolkit! But remember, it's crucial to first confirm that you indeed have an indeterminate form before applying the rule; otherwise, you might end up complicating a problem that's already straightforward. So, with the concept of L'Hôpital's Rule under our belts, we can confidently move forward to applying it to our specific limit problem, simplifying it step by step until we arrive at the solution. Understanding this rule is key to unlocking a wide range of limit problems in calculus, and with practice, it will become second nature.

Applying L'Hôpital's Rule Step-by-Step

Okay, let's roll up our sleeves and apply L'Hôpital's Rule to our limit:

$\lim _{x \rightarrow 0} \frac{4 \cos (7 x)-4}{11 x}$

We've already confirmed that this limit results in the indeterminate form 0/0. Now, the fun begins! First, we need to find the derivatives of the numerator and the denominator separately. Let's start with the numerator, f(x) = 4cos(7x) - 4. The derivative, f'(x), can be found using the chain rule. Remember, the derivative of cos(u) is -sin(u) * u'. So:

f'(x) = 4 * (-sin(7x)) * 7 - 0 = -28sin(7x)

Now, let's tackle the denominator, g(x) = 11x. This one's much simpler! The derivative, g'(x), is just:

g'(x) = 11

Great! We've got our derivatives. Now we can rewrite the limit using L'Hôpital's Rule:

$\lim _{x \rightarrow 0} \frac{4 \cos (7 x)-4}{11 x} = \lim _{x \rightarrow 0} \frac{-28 \sin (7 x)}{11}$

See how we've transformed the limit into something a bit more manageable? The next step is to evaluate this new limit. We're getting closer to the solution! Remember, the key to mastering L'Hôpital's Rule is practice, so let’s keep pushing forward and see how this plays out.

Evaluating the New Limit

Alright, we've arrived at our new limit after applying L'Hôpital's Rule:

$\lim _{x \rightarrow 0} \frac{-28 \sin (7 x)}{11}$

Now, let's try substituting x = 0 directly into this expression:

$\frac{-28 \sin (7 * 0)}{11} = \frac{-28 \sin (0)}{11} = \frac{-28 * 0}{11} = 0$

Voila! The limit is 0. We didn't encounter another indeterminate form this time, so we're in the clear. This demonstrates the power of L'Hôpital's Rule in simplifying complex limits. By taking the derivatives of the numerator and the denominator, we transformed a tricky problem into a straightforward one. The result here not only gives us the numerical value of the limit but also reinforces the concept that as x approaches 0, the original function converges to 0. It's like unwrapping a mathematical puzzle, where each step brings us closer to the final picture. So, we've successfully navigated through the intricacies of L'Hôpital's Rule and arrived at our solution. It's a testament to the elegance and utility of calculus in tackling problems that initially seem daunting. Now, let's summarize our journey and highlight the key takeaways from this exploration.

Summary and Key Takeaways

Let's recap what we've done. We started with the limit:

$\lim _{x \rightarrow 0} \frac{4 \cos (7 x)-4}{11 x}$

We identified that directly substituting x = 0 resulted in the indeterminate form 0/0. This led us to employ L'Hôpital's Rule, a powerful tool for dealing with such cases. We took the derivatives of the numerator and the denominator separately:

• Numerator derivative: -28sin(7x)
• Denominator derivative: 11

This transformed our limit into:

$\lim _{x \rightarrow 0} \frac{-28 \sin (7 x)}{11}$

Finally, we evaluated this new limit by substituting x = 0, which gave us the result 0.

The key takeaway here is that L'Hôpital's Rule is a fantastic technique for evaluating limits that yield indeterminate forms. Remember to always check for an indeterminate form before applying the rule. Also, be meticulous with your derivatives – a small mistake there can throw off the entire solution. Think of L'Hôpital's Rule as a strategic move in the game of calculus, allowing you to simplify problems and find solutions more efficiently. With this example under your belt, you're well-equipped to tackle similar limit problems with confidence. Keep practicing, and you'll become a L'Hôpital's Rule pro in no time! Understanding this rule not only expands your calculus toolkit but also deepens your appreciation for the elegance and power of mathematical problem-solving.