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Limits and cancellation

A limit says what the function approaches, even if it is not defined at the point. The expression 0 divided by 0 is not an answer. You factorise, cancel the common factor and insert. For a fraction of polynomials as $x$ tends to infinity, the highest degree decides. You can check by inserting a large number, but that large number is not itself the limit.

\lim_{x \to a}\frac{x^2 - a^2}{x - a} = 2a

after cancellation

\lim_{x \to \infty}\frac{2x + 1}{x - 4} = 2

degree against degree

Symbols

$a$	the point you approach
$L$	the limit

Example

$\dfrac{x^2 - 9}{x - 3} = x + 3$ when $x \ne 3$ , so the limit at $3$ is $6$ . At $x = 100$ , $\dfrac{201}{96} \approx 2.09$ , close to the limit $2$ .

Cancel only a factor that appears in both the numerator and the denominator, never a single term.

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Part of Mathematics R1: Limits and continuity.