Subject: Two limits How do I find

lim (1-cosx)/(x^2) as x-> 0


lim (tan3x)/x as x->0


Many of the limits in first semester calculus that involve trig functions revolve around one fundamental fact. That fact is

The problem then usually involves some algebraic and/or trigonometric manipulation to use this fact. As is often the case in algebra and trigonometry you need to see a pattern somewhere.
   In your first problem the numerator is 1 - cos x and I know that

1 - cos2 x = sin2 x.

I also know that

1 - cos2 x = (1 - cos x)(1 + cos x)

and hence I can proceed as follows.


In your second problem tan3x = sin3x/cos3x and thus

For the first factor to be of the form sint/t so that you can apply the fact mentioned above you need t = 3x so multiply the numerator and denominator by 3 to get

Now take the limit as x->0.



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