2.5.2-eg-1 - x → 3 agree it suffices to pick a so that 8...

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Example For which value(s) of a is the function f ( x ) = ( x 2 - 1 , for x < 3 ax , for x 3 continuous at x = 3 ? Solution: For continuity at x = 3, we need to find a so that lim x 3 - f ( x ) = lim x 3 + f ( x ) = f (3) . Taking the left-hand limit, lim x 3 - f ( x ) = lim x 3 - x 2 - 1 = 3 2 - 1 = 8 . The right-hand limit satisfies lim x 3 + f ( x ) = lim x 3 + ax = 3 a. In order that the left-hand and right-hand limits as
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Unformatted text preview: x → 3 agree, it suffices to pick a so that 8 = 3 a, or a = 8 / 3 . With this choice we obtain lim x → 3-f ( x ) = lim x → 3 + f ( x ) = f (3) = 8 , and the function is continuous as x = 3....
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