Q.By considering values of $x$ close to $3$, on both sides, determine $\lim_{x\to3}\dfrac{x^2-9}{x-3}$.

The limit of a function $f(x)$ as $x$ approaches a

value $a$ describes the single number that $f(x)$ gets closer and closer to as $x$ is chosen

closer and closer to $a$, on either side, without $x$ ever equalling $a$ itself. It is defined

through two one-sided limits -- the left-hand limit ($x$ approaching $a$ from below) and the

right-hand limit ($x$ approaching $a$ from above) -- and the (two-sided) limit exists exactly

when these two one-sided limits agree; if a function is built from different formulas on either

side of $a$, or involves an expression like $|x|$ that changes sign there, checking the two

one-sided limits separately is the standard first step. Crucially, the limit never depends on

the actual value $f(a)$ (or on whether $f$ is even defined at $a$): a rational function such as

$(x^2-1)/(x-1)$ can have a perfectly well-defined limit at $x=1$ despite being undefined there

itself, once the common factor causing the $0/0$ form is cancelled algebraically.