Meaning and Notation of Limit

The idea of a limit is the foundation on which the whole of calculus — and every business-mathematics application built on it, from continuous compounding to marginal analysis in the very next chapter — rests. Informally, the limit of a function $f(x)$ as $x$ approaches a point $a$ describes the value that $f(x)$ gets arbitrarily close to as $x$ itself gets arbitrarily close to $a$, whether or not $f$ is actually defined at $a$. This is written:

read as "the limit of $f(x)$, as $x$ tends to $a$, equals $L$." The Gujarat Std-12 Statistics (Business Mathematics & Statistics) syllabus introduces limits precisely because they are the gateway both to differentiation and to genuinely useful business models such as continuous compound interest, covered later in this chapter.

Left-hand limit (LHL) and right-hand limit (RHL)

Because $x$ can approach $a$ from below (through values smaller than $a$) or from above (through values larger than $a$), every limit has two one-sided companions:

• Left-hand limit: $\displaystyle\lim_{x \to a^{-}} f(x)$ — the value $f(x)$ approaches as $x$ approaches $a$ through values LESS than $a$.
• Right-hand limit: $\displaystyle\lim_{x \to a^{+}} f(x)$ — the value $f(x)$ approaches as $x$ approaches $a$ through values GREATER than $a$.

Existence of a limit — the golden rule

If the left-hand and right-hand limits are equal, that common value IS the limit. If they differ even slightly, the (two-sided) limit simply does not exist at that point — even if $f(a)$ itself happens to be perfectly well defined. This one rule is what every left-hand/right-hand limit question in this GSHSEB Std-12 chapter is really testing, and it is also exactly the rule that decides whether a function is continuous at a point (Section 4).