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Limit Laws Suppose that c is a constant and the limits lim f(x) and lim g(x) exist. Then

Calculating Limits Using the Limit Laws. Limit Laws Suppose that c is a constant and the limits lim f(x) and lim g(x) exist. Then. x -> a. x -> a. lim [ f(x) + g(x) ] = lim f(x) + lim g(x). 2. lim [ f(x) - g(x) ] = lim f(x) - lim g(x) .

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Limit Laws Suppose that c is a constant and the limits lim f(x) and lim g(x) exist. Then

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

  2. 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 xn = an where n is a positive integer. __ __ 10. lim n x = n awhere nis a positive integer.

  4. __ _____ 11. lim nf(x) = n lim f(x)where nis 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 ( x2 – 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 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 x -> a x -> a

  11. Example Use the Squeeze Theorem to show that lim x2sin( / x) equals to zero. Graph all of the functions. x -> 0 animation

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