You should realize that:
that limit is
Having a ball teaching myself calculus for DSP...
I'm working with derivatives of trig functions. I'm trying to solve the following limit:
My first thought was to just set x to 0, since the equation would still remain valid. This is obviously wrong, when checking the answer in the back of the book, which gives this limit as 1/2. This question immediately follows a proof regarding the limit of sin(x)/x as x approaches zero always being 1, so I thought it might be helpful to look for ways to reduce this to something along those lines, but I'm just not seeing it. Can someone point me in the right direction?
Many thanks,
Brennon
Well, I had gotten far enough to realise that
which is essentially what you've posted. What I don't see, however, is how you take the limit of this as x approaches zero to be . I don't see, unless my following logic here is correct (but it seems fishy to me):
Furthermore, since
it (should?) follow that also,
Thus,
Am I on the right track?
Thanks again,
Brennon
It actually doesn't remain valid. The reason is that goes toward as . So you're left with a term that goes to 0 and a term that goes toward multiplied together... in other words, this is the indeterminate form.
That's why direct substitution does not work.
Yes, you did it correctly.