An abuse of notation and terminology

It has, unfortunately, become customary to use the notation

$$\lim_{x\rightarrow a}$$

to pose a question rather than to stand for the value of the limit. For example, you will see written exercises such as ``Find the following if it exists:''

$$\lim_{x\rightarrow 0} \frac{1}{x}.$$

Strictly speaking, this is nonsense, as the notation in the display was said to stand for the limit, so how can there be any question as to its existence. More properly, the question should be put, ``Does 1/x have a limit as x approaches ?''