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Evaluate the following Limits: limx→0[(5x-1)2x⋅log(1+x)] - Mathematics and Statistics

Sum

Evaluate the following Limits: `lim_(x -> 0)[((5^x - 1)^2)/(x*log(1 + x))]`

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Solution

`lim_(x -> 0)((5^x - 1)^2)/(x*log(1 + x))`

= `lim_(x -> 0) ((5^x - 1)^2/x^2)/((x*log(1 + x))/x^2)   ...[("As"  x -> 0","  x ≠ 0 therefore x^2 ≠ 0),("Divide Numerator and"),("Denominator by " x^2)]`

= `(lim_(x -> 0)((5^x - 1)/x)^2)/(lim_(x -> 0)(log(1 + x))/x`

= `(log 5)^2/1      ...[(lim_(x -> 0) ("a"^x - 1)/x = log "a"","),(lim_(x -> 0) (log(1 + x))/x = 1)]`

= (log 5)2 

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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. (15) | Page 106
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