Hard limit done in 5 minutes. скачать в хорошем качестве

Hard limit done in 5 minutes.

Welcome to the Math Show! In this video, I'll quickly guide you through finding a challenging limit. We'll be focusing on the limit as x approaches zero from the right of (1 + sin(x))^(1/x). Don't worry if it seems intimidating at first glance, I'll break it down step by step. To simplify the expression, we'll utilize the fact that e^(ln(x)) is equal to x. Since e and ln are inverse functions, we can cancel them out and work with the exponent. Remember, these rules work both ways, so we can go back and forth as needed. Let's proceed by rewriting the limit as follows the limit as x approaches zero from the right of e^(ln(1 + sin(x))^1/x). By taking the expression and placing it inside the ln, we cancel out the e and return to step one, restoring the original limit. Now, let's apply the power rule. The 1/x term in the exponent can be brought down and placed in front. Our expression becomes the limit as x approaches zero from the right of e^(1/x * ln(1 + sin(x))). We can simplify this further by placing 1/x under ln, treating it as an exponent. Since the e function is continuous, we can move the limit and place it as an exponent on top of e. Thus, our expression becomes e^(the limit as x approaches zero from the right of ln(1 + sin(x))/x). Next, we need to evaluate the exponent. When we plug in x = 0, we encounter an indeterminate form of 0/0. To handle this situation, we can apply l'Hôpital's rule, which allows us to differentiate the numerator and denominator separately. Continuing from here, we rewrite the exponent as e^(the limit as x approaches zero from the right of d[ln(1 + sin(x))/dx]/d[x]/d[x]). Now, we differentiate ln(1 + sin(x)) using the chain rule and the derivatives of logs and sines. The derivative will be 1/(1 + sin(x)) times the derivative of (1 + sin(x)), which simplifies to cosine(x). Thus, we can rewrite the exponent as cosine(x)/(1 + sin(x)). The derivative of x is just 1. Applying l'Hôpital's rule, we can write the exponent as cosine(x)/(1 + sin(x))/1. The denominator being 1 doesn't change anything, so we can leave it in this form. Here comes the exciting part! We can now plug in x = 0 directly and find the value of the limit. Substituting x = 0, our expression simplifies to e^cosine(0)/(1 + sin(0)). Cosine(0) is equal to 1, and sine(0) is also 0, so our expression becomes e^1/(1 + 0), which further simplifies to e^1. And there you have it! The value of our limit is e. Thank you for watching this insightful video. If you found it helpful, please give it a like and consider subscribing. Your support is greatly appreciated! 🔮📈🎯 Common Internet Questions about Finding Limits 1️⃣ Question How do you find the limit of (1 + sin(x))^(1/x) as x approaches zero from the right? Answer To find this limit, we can simplify the expression using logarithmic properties and apply l'Hôpital's rule. Let's go through the step-by-step process. 2️⃣ Question What is the basic fact used to simplify the expression? Answer The basic Buy a clever and unique math t-shirt: https://rb.gy/rmynnq A humble limit limit as x approaches 0 from the right, using e, ln, L'Hopital and all the cool tricks. Please visit our Merch Stores and help support the spreading of knowledge:) Our T-Shirt Merch: https://my-store-cf0bfb.creator-sprin... Our Amazon Store for Awesome Merch too: https://amzn.to/3OttJgU Amazon Music Free Trial: https://amzn.to/3PzvbzY Amazon Prime Free Trial: https://amzn.to/3PUrmoN Audible Plus Free Trial: https://amzn.to/3RPcrxI Kindle Unlimited Free Trial: https://amzn.to/3yXGZVs Video Game Bestsellers: https://amzn.to/3aZSX9h Are you a fan of our content and want to support us in a tangible way? Why not check out our merchandise? We have a wide range of products, including t-shirts, hoodies, phone cases, stickers, and more, all featuring designs inspired by our brand and message.By purchasing our merchandise, not only will you be showing your support for our work, but you'll also be able to enjoy high-quality, stylish products that you can wear or use in your daily life. And best of all, a portion of the proceeds goes directly towards helping us continue to create and produce the content you love.So what are you waiting for? Head over to our online store now and browse our selection of merchandise. We're sure you'll find something you love.