Maharashtra State BoardHSC Commerce 11th
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Evaluate the following Limits: limx→0[log(4-x)-log(4+x)x] - Mathematics and Statistics

Sum

Evaluate the following Limits: `lim_(x -> 0)[(log(4 - x) - log(4 + x))/x]`

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Solution

`lim_(x -> 0)(log(4 - x) - log(4 + x))/x`

= `lim_(x -> 0) (log[4(1 - x/4)] - log[4(1 + x/4)])/x`

= `lim_(x -> 0)(log4 + log(1 - x/4) - [log4  log(1 + x/4)])/x`

= `lim_(x -> 0) (log(1 - x/4) - log(1 + x/4))/x`

= `lim_(x -> 0)[(log(1 - x/4))/x - (log(1 + x/4))/x]`

= `lim_(x -> 0) (log(1 - x/4))/((-4)(-x/4)) - lim_(x -> 0) (log(1 + x/4))/(4(x/4)`

= `-1/4 lim_(x -> 0) (log(1 - x/4))/(-x/4) - 1/4 lim_(x -> 0) (log(1 + x/4))/(x/4)`

= `-1/4(1) - 1/4(1)     ...[("As"  x -> 0"," x/4 -> 0"," (-x)/4 _> 0),(and lim_(x -> 0) (log(1 + x))/x = 1)]`

= `-1/2`

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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. (18) | Page 106
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