Another method for proving lim x→0 cos x−1/x = 0 Use the half-...

Question

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

Accepted Answer

Hello. In this video, we are going to be verifying the following statement. We are asked if we can use the half-angle identity for cosine to prove that the limit as X approaches 0 of cosine of X minus 1 divided by X is equal to 0. So, in order to verify the statement, we will first start by using the given half angle identity. The half-angle identity for cosine is cosine of X divided by 2 equal to the positive and negative square root. Of 1 plus cosine of X divided by 2. First, let's go ahead and try to solve for a substitution for the limit. We will go ahead and start by squaring both sides of the statement. That will allow us to rewrite the identity as cosine squared of X divided by 2 equal to 1 plus cosine of X divided by 2. Next, we will reorder the statement and multiply both sides of the equation by 2. That will give us 1 plus cosine of X equal to 2 multiplied by cosine squared of X divided by 2. Now that we have this statement, what we want to do is we want to try to rewrite the left hand side in a similar fashion to the numerator of the given limit, cosine X minus 1. In order to do this, let's go ahead and subtract 2 from both sides of the equation. That will allow us to rewrite the identity as cosine of X minus 1, equal to 2 multiplied by cosine squared of X divided by 2 minus 2. Now that we have this statement, how can we replace cosine squared X divided by 2 in a simplified manner? We'll recall that there we have a Pythagorean identity, which states that cosine squared of X is equal to 1 minus squad of X. If we apply this identity to cosine squared of x divided by 2, we can rewrite the identity to be cosine of X minus 1 equal to 2 multiplied. By 1 minus squared of X divided by 2 minus 2. Now we are going to simplify the right-hand side. We will distribute 2 across 1 minus square of X divided by 2, and that will allow us to rewrite the right-hand side of the identity as 2 minus 2 squared of X divided by 2 minus 2. And if we combine two and 2 together, those terms will cancel out, leaving us with cosine of X minus 1 equal to -2 squared of X divided by 2. So what we've done is we created a substitute for the limit, so that we can verify if we can use the half-angle identity to prove that the limit is equal to 0. Let's go ahead and apply this substitute to the original limit. Now recall that the original limit is the limit as X approaches 0 of cosine of X minus 1 divided by X. What we are going to do is we are going to apply our substitution to the numerator of the limit. That will allow us to rewrite the limit, as the limit as X approaches 0 of -2 squared of X divided by 2, all divided by X. Now, how can we continue from this step? Well, in order to continue, let's go ahead and simplify the argument of sine squared. We will go ahead and create a substitution and state that the variable U is equal to X divided by 2. Equally, what this means is that the variable X will equal to 2 U. Furthermore, as X approaches 0, since we want our limit to be in terms of you, that means that you will also approach 0. So by applying these substitutions, we can rewrite the limit to be the limit as you approaches 0 of -2, sin squared. Of You Divided by 2 U. Now let's go ahead and simplify. Since we have a factor of 2 in both the numerator and denominator, we can simplify the limit to be the limit as you approach a 0, of negative sine squared of you divided by U. Next, what we can go ahead and do is we can go ahead and rewrite this as a product of two functions. We can split apart, sign square to view. As the product of two sine of you functions. And so if we apply this rewrite to the limit, we can rewrite the limit as the limit as you approaches zero, multiplied by negative sign of you, multiplied by the function, sign of you divided by you. And if we apply the limit to both of the functions, we will end up having two limits. We have the negative limit as you approaches 0. Of sign of you multiplied by the limit. As you approaches 0, a sign of you. Divided by you. Now, notice that this is in the form of the special sign of view of X function. What this means is that by using the special limit, we can rewrite the limit as you approach a zero of sign of you divided by you as 1. And if we directly substitute 0 into sine of view, this limit is going to simplify to be zero. That will allow us to rewrite the limit as 0, multiplied by 1. Which is going to equal to 0, and that verifies that our original limit is equal to 0. And so the overall solution to this problem is going to be, yes, we can use the half-angle identity to verify that the given limit is equal to 0. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

Key Concepts

Limits

Trigonometric Identities

L'Hôpital's Rule

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