$f\colon [-1,1] \rightarrow \mathbb{R}$ is differentiable. I need to show $\forall \epsilon >0,\ \exists$ a polynomial P s.t. $|f(x)-P(x)|\leq \epsilon|x|$.
I think I need to approximate the difference quotient $\frac{f(x)-f(0)}{x-0}$ with a polynomial but this isn't continuous at $0$ so I can't use Weierstrass to know that such an approximating sequence of polynomials exists.