<s>[INST] Loss of significance error 
Give exact ways of avoiding loss-of-significance errors in the
  following computations:
a. $log(x+1)-log$, with large $x$
b. $\frac{1-cosx}{x^2}$, with $x\approx 0$
c.$(1+x)^{1/3}-1$, with $x\approx 0$

Am I doing this correctly? 
a. I got $$log(x+1)-log(x)= log(\frac{x+1}{x})= log(1+\frac{1}{x}) \approx \frac{1}{x}-\frac{1}{2x^2}+\frac{1}{3x^3} ...$$
c. I got $$\frac{x}{(1+x)^{2/3}+(1+x)^{1/3}+1}$$
b. I am stuck
 [/INST] a) 
$$\log{(x+1)} = \log{x} + \log{\left (1 + \frac{1}{x} \right )}$$
Use the Taylor series approximation for $\log{y}$ for small values of $y$:
$$\log{(x+1)} - \log{x} = \frac{1}{x} - \frac{1}{2 x^2} + \frac{1}{3 x^3} + \dots$$
b) Again, use the Taylor series for $\cos{x}$ about $x=0$:
$$\frac{1-\cos{x}}{x^2} = \frac{1}{2} - \frac{x^2}{24} + \dots$$
c) Taylor expansion or a binomial expansion:
$$\begin{align}(1+x)^{1/3}&= 1 + \frac{1}{1!} \frac{1}{3} x + \frac{1}{2!} \frac{1}{3} \left (-\frac{2}{3} \right ) x^2 + \frac{1}{3!} \frac{1}{3} \left (-\frac{2}{3} \right ) \left (-\frac{5}{3} \right ) x^3 + \ldots \\  \end{align} $$
so that
$$(1+x)^{1/3} - 1 = \frac{1}{3} x - \frac{1}{9} x^2 + \frac{5}{81} x^3 - \ldots$$
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