Limit inferior/superior of a function and its inverse

Question

I'm having a trouble with the veracity of the following statement: Let $f:\mathbb{R}\to\mathbb{R}$ be a function. Assuming that $f$ and $\dfrac{1}{f}$ are bounded in a neighborhood of $a$:$$\limsup_{x\to a^{+}}(f)=\left(\liminf_{x\to a^{+}}\dfrac{1}{f}\right)^{-1}$$ I think it's false but I can't come with a counterexample. In case it's true, can you give me any hint? Thanks in advance