Limit Laws Suppose that c is a constant and the limits

lim f(x) and lim g(x) exist. Thenx -> a x -> a

Calculating Limits Using the Limit Laws

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.

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 xn = an where n is a positive integer.

10. lim n x = n a where n is a positive integer.

____

11. lim n f(x) = n lim f(x) where n is ax -> a

integer. ( If n is even, we assume that

lim f(x) > 0. )x -> a

__ _____

Example Evaluate the limit if exists.

lim ( x2 – x – 12 ) / (x + 3)x -> -3

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

Theorem 1 lim f(x) = L if and only if

lim f(x) = L = lim f(x). x -> a

x -> a - x -> a+

Example Find the limit if exists.

lim ( (1 / x) – (1 / | x |))x -> 0+

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 x -> a

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 x -> a

x -> a

Example Use the Squeeze Theorem to show that lim x2 sin( / x)

equals to zero. Graph all of the functions.

x -> 0

animation