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# Evaluate the following Limits: limx→0[x(6x-3x)(2x-1)⋅log(1+x)] - Mathematics and Statistics

Sum

Evaluate the following Limits: lim_(x -> 0)[(x(6^x - 3^x))/((2^x - 1)*log(1 + x))]

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#### Solution

lim_(x -> 0)(x(6^x - 3^x))/((2^x - 1)*log(1 + x))

= lim_(x -> 0)(x(3^x*2^x - 3^x))/((2^x - 1)*log(1 + x))

= lim_(x -> 0) (x*3^x(2^x - 1))/((2^x - 1)*log(1 + x)

= lim_(x -> 0) (x*3^x)/(log (1 + x))    ...[("As"  x -> 0","  2^x -> 2^0),("i.e." 2^x -> 1  therefore 2^x ≠ 1),(therefore 2^x - 1 ≠ 0)]

= lim_(x -> 0) (3^x)/((log(1 + x))/x

= (lim_(x -> 0) 3^x)/(lim_(x -> 0) (log(1 + x))/x

= 3^0/1       ...[lim_(x -> 0) (log(1 + x))/x = 1]

= 1

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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 7 Limits
Miscellaneous Exercise 7 | Q II. (13) | Page 106
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