Limits

If x approaches 'a' but never

If x approaches 'a' but never equals 'a', and f(x) approaches a specific value 'l', then we write lim(x→a) f(x) = l. The formal ε-δ definition states:

Left

Left-hand limit: lim(x→a⁻) f(x) when x approaches 'a' from values less than 'a'. Right-hand limit: lim(x→a⁺) f(x) when x approaches 'a' from values gr

If lim(x→a) f(x) = l

If lim(x→a) f(x) = l and lim(x→a) g(x) = m, then: lim(x→a)[f(x) ± g(x)] = l ± m, lim(x→a)[f(x) × g(x)] = l × m, lim(x→a)[f(x)/g(x)] = l/m (when m ≠ 0)

When both numerator and denominator approach

When both numerator and denominator approach zero (0/0 form), factor both expressions and cancel common factors. Example: lim(x→1) (x²-1)/(x-1) = lim(

Used when expressions involve square roots

Used when expressions involve square roots or irrational terms. Multiply numerator and denominator by the rationalizing factor. Example: lim(x→0) (√(1