Find k with little oh notation? For which values of $k$ are the following true (as $x \to 0$)?
(a) $\sqrt{1+x^2} = 1 + o(x^k)$
(b) $\sqrt[3]{1+x^2} = 1 + o(x^k)$
(c) $1 - \cos(x^2) = o(x^k)$
(d) $1 - \cos^2 x = o(x^k)$
How would you find $k$ for problems like these?

For (a), have a look at $\sqrt{1+x^2}-1$, which we can transform with the standard trick of multiplying and dividing by the conjugate:
$$ \sqrt{1+x^2}-1=\frac{(\sqrt{1+x^2}-1)(\sqrt{1+x^2}+1)}{\sqrt{1+x^2}+1}=\frac{x^2}{\sqrt{1+x^2}+1}\sim \frac{x^2}2$$
A similar trick helps with (b). For (c) and (d), have a look at the Taylor-expansion of the cosine )as the tagging suggests)
