Thursday, September 14, 2017

Limits with Trig



Is there any way to evaluate
lim
without using L'Hopital's Rule? I was trying to use some of the standard trig limits (e.g. \lim_{x\to 0} \frac{\sin x}{x} = 1), etc. but couldn't figure it out.




Thank you.


Answer



Adding and subtracting x in nummerator we can split the expression under limit as \frac{\cos x-1} {x} +\frac{x-\sin x} {x^2} The first expression tends to 0 because (1-\cos x) /x^2\to 1/2 and the second expression also tends to 0 as shown below.



Since the second expression is an odd function it is sufficient to prove that the expression tends to 0 as x\to 0^{+}. Next note the famous inequality $$\sin x

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f \colon A \rightarrow B and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...