In the question
Question on one-sided derivatives,
it is shown that if $f$ is differentiable on $]x_0,x_0+\delta[$ for some $\delta>0$, such that $\;\lim_{x\rightarrow {x_{0}}^{+}}f^{\prime}(x)\;$ exists, then $$f^{\prime}_{+}(x_{0})=\lim_{\epsilon\rightarrow 0^{+}} \frac{f(x_0+\epsilon)-f(x_0)}{\epsilon}$$ exists, but the converse is not true.
Question: Suppose $f$ is differentiable on $]x_0,x_0+\delta[$ for some $\delta>0$. Can $f^{\prime}_{+}(x_{0})$ exist but the limit $\;\lim_{x\rightarrow {x_{0}}^{+}}(x-x_{0})f^{\prime}(x)\;$ does not exist ?
In other words, does the existence of $f^{\prime}_{+}(x_{0})$ imply the existence of the limit $\;\lim_{x\rightarrow {x_{0}}^{+}}(x-x_{0})f^{\prime}(x)\;$ ?
Counterexample: $x_0=0$, $f(0)=0$, and $$f(x)=x^2\sin(x^{-3})$$ for $x\ne0$.