\begin{align*} f(x) & = P(x \text{changes in} [0,T]) \\ & = \lim_{n \rightarrow \infty} \binom{n}{x} p^x (1-p)^{n-x}\\ & = \lim_{n \rightarrow \infty} \binom{n}{x} (\frac{\lambda T}{n})^x (1-\frac{\lambda T}{n})^{n-x}\\ & = \lim_{n \rightarrow \infty} \frac{n(n-1)...(n-x+1)}{x!} ( \frac{\lambda T}{n})^x (1- \frac{\lambda T}{n})^{n-x}\\ & = \frac{(\lambda T)^x}{x!} \lim_{n \rightarrow \infty} \frac{n(n-1)...(n-x+1)}{n \cdot n \cdot ... \cdot n} (1-\lambda \frac{T}{n})^{n}(1-\lambda \frac{T}{n})^{-x}\\ & = \frac{(\lambda T)^x}{x!} \lim_{n \rightarrow \infty} (1-\frac{1}{n})...(1-\frac{x-1}{n}) (1- \frac{\lambda T}{n})^{n}(1- \frac{\lambda T}{n})^{-x}\\ & = \frac{(\lambda T)^x}{x!} \lim_{n \rightarrow \infty} (1- \frac{\lambda T}{n})^{n} \lim_{n \rightarrow \infty} (1- \frac{\lambda T}{n})^{-x}\\ & = \frac{(\lambda T)^x}{x!} \lim_{n \rightarrow \infty} (1- \frac{\lambda T}{n})^{n} \cdot 1\\ & = \frac{(\lambda T)^x}{x!} e^{-\lambda T} \end{align*}