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Numerade Educator

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Problem 34 Hard Difficulty

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 0} \frac{\cos mx - \cos nx}{x^2} $

Answer

$\lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^{2}}=\left(n^{2}-m^{2}\right) / 2$

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Video Transcript

So for this problem we're gonna be using reptiles rule. Um in order to find the limit as X approaches zero, the function. So in this case what we have is coastline mx minus coastline. An X over X squared. And because it's zero, we know that that's going to give us 0/0. Which is the indeterminant form. So when we take the derivative of the top and the bottom we'll end up getting is a negative mm sign mx plus and sign nx All divided by two x. Then we would still get the indeterminate form. When we evaluated at X equals zero, we would just get 0/0 again. So we have to take the derivative one more time. That's gonna end up giving us uh negative M squared. Uh Co sign annex plus and squared could sign mhm. An axe that's going to end up giving us um just M squared negative M squared plus M squared As a result when we plug in zero. And this is just going to become, too when we took the derivative. So our final answer is going to be n squared minus m squared or negative m squared plus n squared over to um and that will be the final value of the limit. So this problem shows that even if we don't have specific values, we can still evaluate limits um depending on circumstances.