Algebra of Limits and Methods of Evaluation

Once the existence of a limit is settled, evaluating it is made far easier by a small set of algebraic rules. If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ (both limits existing), then:

These rules reduce the limit of a complicated expression to the limits of its simpler pieces — but they only apply once each individual limit is known to exist, which is where the three practical evaluation methods below come in.

1. Direct substitution. If $f$ is a polynomial, or a rational function whose denominator does not vanish at $x=a$, the limit is simply the value obtained by substituting $x=a$ directly: $\lim_{x \to a} f(x) = f(a)$ This works because polynomials and well-behaved rational functions are continuous wherever they are defined (made precise later in this chapter) — there is no gap between the tendency near $a$ and the value at $a$.

2. Factorisation. If direct substitution gives the meaningless form $\dfrac{0}{0}$, both numerator and denominator share a common factor that vanishes at $x=a$ (typically $(x-a)$). Factorise both, cancel the common factor — valid because a limit only ever considers $x$ near $a$, never $x=a$ itself — and then substitute. …