Step in Spivak's proof that derivative of a $f: \mathbb{R^n}\rightarrow\mathbb{R^m}$ is unique

Question

On page 16 of Spivak's Calculus on Manifolds,Spivak proves the derivative of a function $f: \mathbb{R^n}\rightarrow\mathbb{R^m}$ is unique if it exists. The last line of the proof is $$0=lim_{t\to{0}}\frac{|\lambda{(tx)}-\mu{(tx)}|}{|tx|}=\frac{|\lambda{(x)}-\mu{(x)}|}{|x|},$$ but I don't understand why $$lim_{t\to{0}}\frac{|\lambda{(tx)}-\mu{(tx)}|}{|tx|}=\frac{|\lambda{(x)}-\mu{(x)}|}{|x|}.$$

Does anyone know why? Is it because $|t|$ factors out of the numerator and denominator?

Answer 1

You omitted important parts of the proof preceding that statement. Before the statement, Spivak proves that $\lim_{h\to 0}\frac{\lambda(h)-\mu(h)}{|h|}=0$. And as you recognized, we can factor out $|t|$ on both sides. Hence, since $x\neq 0$ in this case, we must have $|\lambda(x)-\mu(x)|=0$.