Alternative definition of derivative in real analysis

I am working on the following definition of derivative.

Let $I$ be an open interval and $f \colon I \rightarrow \mathbb{R}$ be a function, and let $x \in I.$ The function $f$ is differentiable at $x$ if and only if $\exists$$f^{\prime}(x) \in \mathbb{R}$ and a function $r \colon I \rightarrow \mathbb{R}$ satisfying$$\lim_{y \rightarrow x} \dfrac{r(y)}{y-x}=0$$ such that for all $y \in I$,$$f(y)=f^{\prime}(x)(y-x)+f(x)+r(y).$$

I understand the definition very well. But I have the following question.

Why is the condition $\lim_{y \rightarrow x} \dfrac{r(y)}{y-x}=0$ needed? Can you convince me that why the condition $\lim_{y \rightarrow x} r(y)=0$ is not sufficient? I know that the first condition is stronger, but I was wondering why the second one is not good enough.

Any help is appreciated! Thank you in advance!