¿Cómo resolver el problema definitivo en matemáticas para el examen de ingreso a posgrado?

x- gt; 0

Numerador

√(1 x^2)= 1 (1/2)x^2 o(x^2)

x √(1 x^2)= 1 x (1/2)x^2 o(x^2)

ln[x √(1 x^2)]

= ln[1 x (1/2)x^2 o(x^2)]

=[x (1/2)x^2)]-(1/2)[x (1 /2)x^2]^2 o(x^2)

=[x (1/2)x^2)-(1/2)[x^2 o(x^2) ] o(x^2)

= x o(x^2)

ln(1 x)= x -(1/2)x^2 o(x^2)

ln(1 x)-ln[x √(1 x^2)]=-(1/2)x^2 o(x^2)?

Denominador

ln(x √(1 x^2))= ln(1 x o(x))= x o(x)

ln(1 x) = x o(x)

ln(1 x). ? ln(x √(1 x^2)) =x^2 o(x^2)

//

lim(x->;0)[1/ln( x √(1 x^2))-1/ln(1 x)]

= lim(x->;0)[ln(1 x)-ln(x √(1 x^2 ]]/[ln(x √(1 x^2)).ln(1 x) ]

= lim(x->;0) -(1/2)x^2/x^ 2

=-1/2

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