Assume we have a function: $$ f(x) = (1+x)^{k} $$ with a fixed integer $k$. Next, let us approximate $f(x)$ by Taylor series around $0$: $$ f(x) = 1 + kx + \frac{1}{2}k(k-1)x^{2} + o(x^{2}). $$
Next, let us take some fixed small $x$. Would it make sense to consider the function $f$ as a function of $k$ and write $$ f(k) \approx 1 + (x-\frac{1}{2}x^{2})k + \frac{1}{2}x^{2}k^{2} $$ ?
PS I know that there is multivariate version of Taylor series, I am just confused with this approach.