Evaluating Limits: Algebra of Limits and Indeterminate Forms

Once the meaning of a limit is clear, the practical skill this chapter builds is evaluating limits without drawing a graph or plugging in a long table of values every time. Gujarat board Std 12 Statistics numericals on limits fall into two broad situations.

(A) Algebra of Limits — combining limits directly

If $\displaystyle\lim_{x\to a} f(x) = L$ and $\displaystyle\lim_{x\to a} g(x) = M$ both exist, then:

• Sum/Difference: $\displaystyle\lim_{x\to a}[f(x)\pm g(x)] = L \pm M$
• Constant multiple: $\displaystyle\lim_{x\to a}[k\cdot f(x)] = k\,L$
• Product: $\displaystyle\lim_{x\to a}[f(x)\cdot g(x)] = L \cdot M$
• Quotient: $\displaystyle\lim_{x\to a}\dfrac{f(x)}{g(x)} = \dfrac{L}{M}$, provided $M \neq 0$

These rules let the limit of a complicated expression be broken into limits of its simpler pieces — but they only apply once each individual piece's limit is already known, which is why substituting $x=a$ directly is always tried first.

(B) Indeterminate forms — when direct substitution fails

Substituting $x=a$ directly sometimes produces a meaningless form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ — this does not mean the limit fails to exist, only that the expression must first be simplified before substituting.

• $\frac{0}{0}$ form, a polynomial/algebraic ratio: factorise the numerator and denominator and cancel the common factor causing both to vanish at $x=a$, then substitute.
• $\frac{0}{0}$ form, an expression with a surd: rationalise the numerator or denominator (multiply by the conjugate) first, then cancel and substitute. …