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

Question

• Evaluate lim   ⟨   1 - log(1 + x)⟩

                  x→0         x         x²

Answer 1

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

Answer 2

Use l'Hospital's Rule twice:

 

lim_x→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