Key expansions around x = 0 (valid for |x| < relevant radius of convergence):

sin(x) = x - $x^3$/3! + $x^5$/5! - ... cos(x) = 1 - $x^2$/2! + $x^4$/4! - ... tan(x) = x + $x^3$/3 + 2$x^5$/15 + ... $e^x$ = 1 + x + $x^2$/2! + $x^3$/3! + ... ln(1+x) = x - $x^2$/2 + $x^3$/3 - $x^4$/4 + ... (|x| < 1) (1+x)^n = 1 + nx + n(n-1)$x^2$/2! + ... (|x| < 1, binomial series) sin^(-1)(x) = x + $x^3$/6 + 3$x^5$/40 + ... tan^(-1)(x) = x - $x^3$/3 + $x^5$/5 - ...

How to use in limits: For lim(x->0) (sin x - x + $x^3$/6) / $x^5$: Replace sin x = x - $x^3$/6 + $x^5$/120 - ... Numerator = (x - $x^3$/6 + $x^5$/120 - ...) - x + $x^3$/6 = $x^5$/120 - ... Divide by $x^5$: limit = 1/120.

JEE Tip: Keep only enough terms to cancel. If substituting the first term gives 0/0, include one more term from each expansion.