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The limit as x approaches what? Goodness.
There are different ways to approach this:
1. Since both numerator and denominator evaluate to zero as x->0, l'H˘pital's rule applies.
Differentiate the top with respect to x to get:
and the bottom to get
As x->0, both numerator and denominator still -> 0, thus we can apply again the rule, and differentiate:
As x->0, the sin(x) term vanishes, and the cos(x) terms cancel out, resulting in -4 over 2 in the denominator.
So the limit is -2.
2. If you have done series expansions before, expand numerator into a power series, taking only terms up to x^4:
Dividing by the denominator leaves us with
and as x->0,
-1152/576 = -2 as before.
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