1. Limit Laws Suppose that c is a constant and the limits lim f(x) and lim g(x) exist. Then x -> a Calculating Limits Using the Limit Laws

2. 1.lim [ f(x) + g(x)] = lim f(x) + lim g(x) 2. lim [ f(x) - g(x)] = lim f(x) - lim g(x) 3. lim [ c f(x)] = c lim f(x) 4. lim [ f(x) g(x)] = lim f(x) lim g(x) 5. lim [ f(x)] / [g(x)] = lim f(x) / lim g(x) where lim g(x) is not equal to zero.

3. 6. lim [ f(x)] n = [ lim f(x) ] n where n is a positive integer. 7. lim c = c 8. lim x = a 9. lim x n = a n where n is a positive integer. 10. lim n  x = n  a where n is a positive integer. __

4. 11. lim n  f(x) = n  lim f(x) where n is a x -> a integer. ( If n is even, we assume that lim f(x) > 0. ) x -> a __ _____

5. Example Evaluate the limit if exists. lim ( x 2 – x – 12 ) / (x + 3) x -> -3

6. If f is a polynomial or a rational function and a is in the domain of f, then lim f(x) = f(a). x -> a

7. Theorem 1 lim f(x) = L if and only if lim f(x) = L = lim f(x). x -> a x -> a - x -> a +

8. Example Find the limit if exists. lim ( (1 / x) – (1 / | x |)) x -> 0 +

9. Theorem 2 If f(x) <= g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then lim f(x) <= lim g(x). x -> a

10. The Squeeze Theorem If f(x) <= g(x) <= h(x) when x is near a (except possibly at a) and lim f(x) = lim h(x) = L, then lim g(x) = L. x -> a

11. Example Use the Squeeze Theorem to show that lim x 2 sin(  / x) equals to zero. Graph all of the functions. x -> 0 animation