Question

The exhaustive interval of x for which ln(1 + x) < x is


(0, 1)


(0, ∞)


(–1, ∞) – {0}


(1, ∞)

Answer 1

Answer:

The largest set of real values of ′x′ for which ln(1+x)≤x is. A. (−1,∞). B. (−1,0)∪(0,∞). C. [0,∞). D. (0,∞). Answer. Correct option is. C. [0,∞).

Answer 2

Given : ln(1 + x) < x

To find : The exhaustive interval of x

Solution:

ln(1 + x) < x

=> ln(1 + x) -  x  < 0  

or x - ln(1+x) > 0

ln(1 + x)  = x - x²/2  + x³/3 - x⁴/4 + ____

=>  ln(1 + x) -  x  =   - x²/2  + x³/3 - x⁴/4 + ____

 ln(1 + x) is defined if  1 + x > 0  => x > - 1

 ln(1 + x) -  x  =   - x²/2  + x³/3 - x⁴/4 + ____

if x = 0

Then RHS is 0

Hence   ln(1 + x) -  x = 0 for x = 0

for -1 < x < 0

RHS is -ve  as all term becomes negative

for x  > 0

also RHS is -ve      as  magnitude of terms is decreasing and initial term is -ve

Hence  The exhaustive interval of x for which ln(1 + x) < x is

(–1, ∞) – {0}

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