Where do I even start with this one?
lim x -> 0 of ( (2 - cos(3x) - cos(4x) ) / x )
Is there some identity that I can use that I'm not seeing.. ? I can't multiply by a conjugate can I?
The slacker's way is to use l'Hopital's Rule.
Otherwise you could substitute the Maclaurin series for cos(3x) and cos(4x), simplify the numerator and then cancel the common factor of x. Then take the limit.
Alternatively, you could probably do a clever re-arrangement and get standard forms whose limits are well known. But why hoe the hard road?
If you were going to be a slacker (as I am ), you would notice that the limit tends to $\displaystyle \frac{0}{0}$ by direct substitution.
So you can use L'Hospital's Rule.
$\displaystyle \lim_{x \to 0}\frac{2 - \cos{3x} - \cos{4x}}{x} = \lim_{x \to 0}\frac{\frac{d}{dx}(2 - \cos{3x} - \cos{4x})}{\frac{d}{dx}(x)}$
$\displaystyle = \lim_{x \to 0}\frac{3\sin{3x} + 4\sin{4x}}{1}$
$\displaystyle = \frac{3\cdot 0 + 4\cdot 0}{1}$
$\displaystyle = 0$, as shown previously.