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Evaluate lim 1/x - log(1 +x)/x2

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2 Answers

For small values of x     ln(1+x)~x-x2/2+x3/3-...
1/x-(ln(1+x))/x2=1/x-(x-x2/2+x3/3-...)/x2=1/x-1/x+1/2+x/3-...which =1/2 as x→0
Use l'Hospital's Rule twice:
 
limx→0 1/x - ln(1+x)/x²
= lim (x - ln(1+x))/x²
= lim (1- 1/(x+1) )/2x
= lim (x/(2x+2x²))
= lim (1/(2+4x))
= 1/2