# Hyperbolic sine and landau notation

I have given a function $f$:

$$f = \begin{pmatrix} \sinh(x_1 x_2) \\ \cosh(x_1 x_2) \end{pmatrix}$$

We have to show that this is possible:

$$f = \begin{pmatrix} 2x_1 x_2 \\ 1 \end{pmatrix} + \mathcal O \left( \|x\|^3 \right)$$

We have used the exponential defininition of the hyperbolic function, but we always end up with terms that grow with $\sum_n \frac 1{n!}(x_1 x_2)^n$, which seem to be more than the given $\mathcal O \left( \|x\|^3 \right)$.

We are not sure how to get from $\|x\|^3$ to $x_1 x_2$. How do we attack this problem?

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You are taking Big-O in which direction? The statement you want to show is only true as $x\to 0$, not when $\|x\|\to\infty$.
Note that strictly speaking the use of Landau notation requires specifying "as $x\to a$" for the limit $a$ in which you are evaluating. Sometimes it is clear from context what $a$ is (usually $0$ or $\infty$), but as you see from your question, it is generally much more clear if the limit $a$ is specified explicitly. –  Willie Wong May 23 '12 at 9:09
The problem does not say anything. But since we got it right with $a = 0$, I guess that is meant … –  queueoverflow May 23 '12 at 9:24