Chapter 2 | Limits
161
Theorem 2.5: Limit Laws Let f ( x ) and g ( x ) be defined for all x ≠ a over some open interval containing a . Assume that L and M are real numbers such that lim x → a f ( x ) = L and lim x → a g ( x ) = M . Let c be a constant. Then, each of the following statements holds: Sum law for limits : lim x → a ⎛ ⎝ f ( x )+ g ( x ) ⎞ ⎠ = lim x → a f ( x )+ lim x → a g ( x ) = L + M Difference law for limits : lim x → a ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ = lim x → a f ( x )− lim x → a g ( x ) = L − M Constant multiple law for limits : lim x → a cf ( x ) = c · lim x → a f ( x ) = cL Product law for limits : lim x → a ⎛ ⎝ f ( x ) · g ( x ) ⎞ ⎠ = lim x → a f ( x ) · lim x → a g ( x ) = L · M Quotient law for limits : lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M for M ≠0 Power law for limits : lim x → a ⎛ ⎝ f ( x ) ⎞ ⎠ n = ⎛ ⎝ lim x → a f ( x ) ⎞ ⎠ n = L n for every positive integer n .
= L n
f ( x ) n = lim x → a n
for all L if n is odd and for L ≥0 if n is even and f ⎛ ⎝ x ⎞
f ( x )
Root law for limits : lim x → a
⎠ ≥0 .
We now practice applying these limit laws to evaluate a limit.
Example 2.14 Evaluating a Limit Using Limit Laws
(4 x +2).
Use the limit laws to evaluate lim x →−3
Solution Let’s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. lim x →−3 (4 x +2) = lim x →−3 4 x + lim x →−3 2 Apply the sum law. =4· lim x →−3 x + lim x →−3 2 Apply the constant multiple law. =4·(−3)+2=−10. Apply the basic limit results and simplify.
Example 2.15 Using Limit Laws Repeatedly
2 x 2 −3 x +1 x 3 +4 .
Use the limit laws to evaluate lim x →2
Solution
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