Explicație pas cu pas:
1.a.
[tex]lim \frac{ {x}^{2} - x }{ {x}^{2} - 1 } = lim \frac{x(x - 1)}{(x - 1)(x + 1)} = lim \frac{x}{x + 1} = \frac{1}{1 + 1} = \frac{1}{2} [/tex]
b.
[tex]lim \frac{ ln(x) }{ {x}^{2} } = lim \frac{ \frac{1}{x} }{ {x}^{2} } = lim \frac{1}{2 {x}^{2} } = \frac{1}{2} \times \frac{lim(1)}{lim( {x}^{2}) } = \frac{1}{2} \times \frac{1}{ \infty } = 0[/tex]
2.a.
[tex]f'(x)=( {x}^{3} - 3x + 2 )' = 3 {x}^{2} - 3 = 3( {x}^{2} - 1) = 3(x - 1)(x + 1) [/tex]
b.
[tex]f'(x)=0 = > 3(x - 1)(x + 1) = 0 \\ = > x = 1 \: si \: x = - 1[/tex]
c.
[tex]f( - 1) = 4 = > maxim \\ f(1) = 0 = > minin[/tex]
[tex]crescatoare: \: - \infty < x < - 1 \\ descrescatoare: \: - 1 < x < 1 \\ crescatoare: \: 1 < x < \infty [/tex]
