_ x 0 e ^ x^2 - 3 x x (1+2 x) =

Step-by-step solution

The limit _ x 0 e^ x^2 - 3x x (1 + 2x) is of the indeterminate form 0 0 as x 0 . Use standard approximations near zero: e^u - 1 u , 1 - u u^2 2 , u u , and (1 + u) u . Rewrite the numerator as (e^ x^2 - 1) - ( 3x - 1) = (e^ x^2 - 1) + (1 - 3x) x^2 + (3x)^2 2 = x^2 + 9x^2 2 = 11x^2 2 . The denominator approximates to (x)(2x) = 2x^2 . Substitute into the limit: _ x 0 11x^2 2 2x^2 = 11 4 The limit evaluates to 11 4 .