Q.392. Question on Lagrange's Mean value theorem.

Question 392.
Prove that 0 < [ln (1+x) ]^-1 - x^-1 < 1 for all x > 0 using Mean Value Theorem.

Answer 392.
Let f (x) = ln (1 + x)
=> by MVT for [0, x],
[ln (1 + x) - ln1] / (x - 0) = 1/(1 + c), 0 < c < x
=> c = x/ln(1 + x) - 1
=> 0 < c = x/ln(1 + x) - 1 < x
=> 0 < x/ln(1 + x) - 1 < x
=> 0 < [x - ln(1 + x)] / xln(1 + x) < 1
=> 0 < [ln(1 + x)]^-1 - x^-1 < 1.

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