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Calculating Limits

# Calculating Limits - Calculating Limits Theorem Suppose lim...

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Calculating Limits Theorem. Suppose lim x a f ( x )= L and lim x a g ( x )= M .Le t c be a constant. Then 1. lim x a ± f ( x )+ g ( x ) ² = L + M 2. lim x a ± f ( x ) g ( x ) ² = L M 3. lim x a ± cf ( x ) ² = cL 4. lim x a ± f ( x ) g ( x ) ² = LM 5. lim x a f ( x ) g ( x ) = L M provided M 6 =0. For example, if lim x 5 f ( x ) = 7 and lim x 5 g ( x ) = 8 then lim x 5 f ( x ) g ( x ) = 7 8 . W ecanu sethe se theorems to compute many limits. Example. It is not hard to believe that lim x 4 x = 4 or more generally that lim x a x = a .(The closer x gets to a , the closer x gets to a .) So by the theorem, lim x 4 x 2 = 16, lim x 4 x 3 = 64, etc. More generally, lim x a x n = a n . So by the theorem, if c is any constant, lim x a cx n = ca n . E.g., lim x 4 5 x 3 =5(4 3 ) = 320 . So we can handle constant multiples of powers of x . This means by the theorem we can take limits of polynomials . Example: lim x 4 ( 5

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Calculating Limits - Calculating Limits Theorem Suppose lim...

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