121616949-math.46

# 121616949-math.46 - This often means that it is possible to...

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32 Chapter 2 Instantaneous Rate Of Change: The Derivative This is just what we needed, so by the official definition, lim x a f ( x ) g ( x ) = LM . A handful of such theorems give us the tools to compute many limits without explicitly working with the definition of limit. THEOREM 2.7 Suppose that lim x a f ( x ) = L and lim x a g ( x ) = M and k is some constant. Then lim x a kf ( x ) = k lim x a f ( x ) = kL lim x a ( f ( x ) + g ( x )) = lim x a f ( x ) + lim x a g ( x ) = L + M lim x a ( f ( x ) - g ( x )) = lim x a f ( x ) - lim x a g ( x ) = L - M lim x a ( f ( x ) g ( x )) = lim x a f ( x ) · lim x a g ( x ) = LM lim x a f ( x ) g ( x ) = lim x a f ( x ) lim x a g ( x ) = L M , if M is not 0 Roughly speaking, these rules say that to compute the limit of an algebraic expression, it is enough to compute the limits of the “innermost bits” and then combine these limits.
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Unformatted text preview: This often means that it is possible to simply plug in a value for the variable, since lim x → a x = a . EXAMPLE 2.8 Compute lim x → 1 x 2-3 x + 5 x-2 . If we apply the theorem in all its gory detail, we get lim x → 1 x 2-3 x + 5 x-2 = lim x → 1 ( x 2-3 x + 5) lim x → 1 ( x-2) = (lim x → 1 x 2 )-(lim x → 1 3 x ) + (lim x → 1 5) (lim x → 1 x )-(lim x → 1 2) = (lim x → 1 x ) 2-3(lim x → 1 x ) + 5 (lim x → 1 x )-2 = 1 2-3 · 1 + 5 1-2 = 1-3 + 5-1 =-3...
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