1. lim(x->0) sin$\frac{x}{x}$ = 1 [x must be in radians]
2. lim(x->0) tan$\frac{x}{x}$ = 1
3. lim(x->0) $\frac{1 - cos x}{x}$^2 = 1/2
4. lim(x->0) sin^(-1)$\frac{x}{x}$ = 1
5. lim(x->0) tan^(-1)$\frac{x}{x}$ = 1
6. lim(x->0) $\frac{e^x - 1}{x}$ = 1
7. lim(x->0) $\frac{a^x - 1}{x}$ = ln(a), where a > 0, a != 1
8. lim(x->0) ln$\frac{1 + x}{x}$ = 1
9. lim(x->0) (1 + x)^$\frac{1}{x}$ = e
10. lim(x->infinity) (1 + 1/x)^x = e
11. lim(x->a) $\frac{x^n - a^n}{(x - a)}$ = n * a^(n-1) for all real n
12. lim(x->0) (1 + kx)^$\frac{1}{x}$ = $e^k$

Generalized forms:

• lim(x->0) sin(f(x))/f(x) = 1, provided f(x)->0
• lim(x->0) (e^(f(x)) - 1)/f(x) = 1, provided f(x)->0
• lim(x->0) [f(x)]^(g(x)) when f->1, g->infinity: use e^(lim g(x)(f(x)-1))