17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)

Verified step by step guidance

1. First, substitute x = -1 into the function to check if the limit results in an indeterminate form like 0/0 or ∞/∞. This will help us determine if l'Hôpital's Rule is applicable.
2. After substitution, if the limit is in an indeterminate form, apply l'Hôpital's Rule. This involves taking the derivative of the numerator and the derivative of the denominator separately.
3. Calculate the derivative of the numerator, which is x³ - x² - 5x - 3. The derivative is 3x² - 2x - 5.
4. Calculate the derivative of the denominator, which is x⁴ + 2x³ - x² - 4x - 2. The derivative is 4x³ + 6x² - 2x - 4.
5. Substitute x = -1 into the new function formed by the derivatives to evaluate the limit. If necessary, repeat the application of l'Hôpital's Rule if the result is still an indeterminate form.