Domain and accumulation points

Before even saying “what $f$ does near $x_0$”, we must know when it makes sense to speak of a limit at $x_0$. Two tools are needed: the neighbourhood of a point and the accumulation point of a set. A limit can be taken only at the accumulation points of the domain, because there — and only there — are there infinitely many values of the function nearby to “look at”.

• Neighbourhood of a point
• Accumulation point
• A linguistic reading of the definition
• Isolated point and membership of the domain
• Examples of accumulation points
• When the limit makes sense
• Accumulation points at infinity