Calculus Cheat Sheet Limits Definitions Precise Definition : We say lim f (x) = L if Limit at Infinity : We say lim f (x) = L if we 1700 x->a for every & > 0 there is a 8 > 0 such that whenever 0 <|x -a| < 8 then f (x)-L <8. can make f (x) as close to L as we want by taking x large enough and positive. "Working" Definition : We say lim f (x) = L x->a if we can make f (x) as close to L as we want by taking x sufficiently close to a (on either side of a) without letting x = a . Right hand limit : lim f (x)=L. This has x->a the same definition as the limit except it requires x > a . Left hand limit : lim f (x) =L. This has the x->a same definition as the limit except it requires x<a. There is a similar definition for lim f (x)= L x >-00 except we require x large and negative. Infinite Limit : We say lim f (x) = oo if we x->a can make f (x) arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting x = a. There is a similar definition for lim f (x) =- 00 x->a except we make f (x) arbitrarily large and negative. Relationship between the limit and one-sided limits lim f (x) =L => lim f(x) = lim f (x) = L x->a x->a x->a lim f(x)= lim f (x) => lim f (x) Does Not Exist lim f (x) = lim f (x)= L => lim f (x) = L x->a x->a x->a x->a x->a x->a Properties Assume lim f (x) and lim g (x) both exist and c is any number then, x >a x->a 1. lim[ cf (x)] =c lim f (x) 2. lim[f(x)+g(x)] = lim f (x) + limg (x) 3. lim f(x) g(x) = lim f (x) limg (x) x->a x->a x->a f(x) lim f (x) 4. lim x->a x->a g (x) lim g (x) x->a x-> a provided lim g(x) ¥0 5. lim f(x)]' = lim f (x) x->a x->a n 6. lim x->a if (x ) = n lim f (x) 1 x->a Basic Limit Evaluations at + oo Note : sgn(a)=1 if a>0 and sgn(a) =- 1 ifa<0. 1. lime" = 0 & lim ex = 0 x->00 x>- 00 2. lim ln (x)=0 & lim ln(x) =- 00 x x 0 3. If r > 0 then lim b =0 x 100 x1 4. If r > 0 and x' is real for negative x then lim =0 b x->-00 x' 5. n even : lim x" = 00 x>+00 6. nodd : lim x" = 0 & lim x" = - 00 x>00 x>- 00 7. neven : lim ax" + · · · + b x + c = sgn(a)o x>+00 8. nodd : limax"