Question

Suppose that f is continuous on (-∞, ∞), and that

Answer 1

f(x)=\[ \frac{ sinx-x }{ x^3 } , x \neq0\]
Use the power series expansion for f to find f (0).

Answer 2

Please look up the power series for sin x. Note that we're interested here in the "center" a=0 (which means we want the Maclaurin series for sin x).

Now subtract x from that series.

Now divide the resulting series by x^2.

Finally, let x = 0. If everything works out right, we won't have division by zero and will obtain a numeric value which represents f(0).

Answer 3

I got 0/0?

Answer 4

Do you happen to have handy a reference that gives the Maclaurin series for sin x?

I'd suggest looking up that series, or deriving it from scratch.

Next, subtract x from that series. I get (x^3)/3! + (x^5)/5! - (x^7)/7! +...

Lastly, divide this last result by x^2 and let x = 0 in the resulting quotient.