The Concept of a Limit

A limit describes the single value that a function $f(x)$ approaches as its input $x$ approaches some fixed number $a$ — without any concern for what happens exactly at $x=a$. This distinction is the whole point of the idea: a limit asks about the neighbourhood around $a$, never about the point $a$ itself.

We write

to mean: as $x$ takes values closer and closer to $a$ (on both sides), the outputs $f(x)$ get closer and closer to the single number $L$.

Illustration — value at the point is irrelevant. Consider $f(x) = \dfrac{x^2 - 4}{x - 2}$. At $x = 2$ the formula gives $\dfrac{0}{0}$, which is undefined, so $f(2)$ does not exist. Yet for every $x \neq 2$ we may cancel the common factor: $\dfrac{x^2-4}{x-2} = \dfrac{(x-2)(x+2)}{x-2} = x+2$. As $x$ approaches $2$, $x+2$ approaches $4$. Hence

even though the function is undefined at $x=2$. The limit sees the trend, not the hole.

Illustration — a limit can differ from the function's value. Suppose $g(x) = x + 1$ for all $x \neq 3$, but $g(3) = 10$ (defined separately). Approaching $x = 3$, the outputs follow $x+1 \to 4$, so $\lim_{x\to 3} g(x) = 4$, which is not equal to $g(3) = 10$. The limit reports where the function is heading, which need not match where it has been forced to sit.

Maharashtra's Std XI Commerce Mathematics and Statistics syllabus draws on the same standard principles of limits and calculus taught across Indian higher-secondary and commerce-mathematics curricula — the definition, one-sided limits, the algebra of limits, and the standard limit results developed in the rest of this chapter.