\begin{tabular}{l} 2. Hallar el siguiente limite. \ $$ \qquad \lim _{x ightarrow-2 / 5} \frac{2 x^{3}-3 x^{2}+4 x-5}{x-x^{2}-x^{3}} $$ \ \hline\end{tabular}

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\begin{tabular}{l} 2. Hallar el siguiente limite. \ $$ \qquad \lim _{x ightarrow-2 / 5} \frac{2 x^{3}-3 x^{2}+4 x-5}{x-x^{2}-x^{3}} $$ \ \hline\end{tabular}

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Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow -\frac{2}{5}}\left(\frac{2x^{3}-3x^{2}+4x-5}{x-x^{2}-x^{3}}\right)$$
- step1: Rewrite the expression:
  $$\frac{\lim _{x\rightarrow -\frac{2}{5}}\left(2x^{3}-3x^{2}+4x-5\right)}{\lim _{x\rightarrow -\frac{2}{5}}\left(x-x^{2}-x^{3}\right)}$$
- step2: Calculate:
  $$\frac{-\frac{901}{125}}{\lim _{x\rightarrow -\frac{2}{5}}\left(x-x^{2}-x^{3}\right)}$$
- step3: Calculate:
  $$\frac{-\frac{901}{125}}{-\frac{62}{125}}$$
- step4: Calculate:
  $$\frac{901}{62}$$
Para hallar el límite $$ \lim_{x \rightarrow -\frac{2}{5}} \frac{2 x^{3}-3 x^{2}+4 x-5}{x-x^{2}-x^{3}} $$, primero evaluamos el numerador y el denominador en el punto $$ x = -\frac{2}{5} $$.

1. **Numerador**: 
   $$
   2 \left(-\frac{2}{5}\right)^{3} - 3 \left(-\frac{2}{5}\right)^{2} + 4 \left(-\frac{2}{5}\right) - 5
   $$
   Calculamos cada término:
   - $$ 2 \left(-\frac{2}{5}\right)^{3} = 2 \left(-\frac{8}{125}\right) = -\frac{16}{125} $$
   - $$ -3 \left(-\frac{2}{5}\right)^{2} = -3 \left(\frac{4}{25}\right) = -\frac{12}{25} = -\frac{60}{125} $$
   - $$ 4 \left(-\frac{2}{5}\right) = -\frac{8}{5} = -\frac{200}{125} $$
   - $$ -5 = -\frac{625}{125} $$

Sumando todos los términos:
   $$
   -\frac{16}{125} - \frac{60}{125} - \frac{200}{125} - \frac{625}{125} = -\frac{901}{125}
   $$

2. **Denominador**:
   $$
   -\frac{2}{5} - \left(-\frac{2}{5}\right)^{2} - \left(-\frac{2}{5}\right)^{3}
   $$
   Calculamos cada término:
   - $$ -\frac{2}{5} = -\frac{2}{5} $$
   - $$ -\left(-\frac{2}{5}\right)^{2} = -\frac{4}{25} = -\frac{20}{125} $$
   - $$ -\left(-\frac{2}{5}\right)^{3} = -\left(-\frac{8}{125}\right) = \frac{8}{125} $$

Sumando todos los términos:
   $$
   -\frac{2}{5} - \frac{20}{125} + \frac{8}{125} = -\frac{50}{125} - \frac{20}{125} + \frac{8}{125} = -\frac{62}{125}
   $$

3. **Límite**:
   Ahora, sustituimos los resultados en el límite:
   $$
   \lim_{x \rightarrow -\frac{2}{5}} \frac{-\frac{901}{125}}{-\frac{62}{125}} = \frac{901}{62}
   $$

Por lo tanto, el resultado del límite es:
$$
\frac{901}{62} \quad \text{o} \quad 14 \frac{33}{62}
$$
El límite es $$ \frac{901}{62} $$ o $$ 14 \frac{33}{62} $$.

\begin{tabular}{l} 2. Hallar el siguiente limite. \\ \( \qquad \lim _{x \rightarrow-2 / 5} \frac{2 x^{3}-3 x^{2}+4 x-5}{x-x^{2}-x^{3}} \) \\ \hline\end{tabular}

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