Answer
$-4$
Work Step by Step
To find the $\lim\limits_{x \to -1}\frac{x^3 - x^2 - 5x - 3}{(x + 1)^2}$
when we apply limit directly we get the indeterminate form.
So we calculate the given limit as follows
First, factorize the numerator:
\[ x^3 - x^2 - 5x - 3 = (x + 1)(x^2 - 2x - 3) = (x + 1)(x - 3)(x + 1) \]
Now, the expression becomes:
\[ \lim_{x \to -1} \frac{(x + 1)(x - 3)(x + 1)}{(x + 1)^2} \]
Cancel out the common factor \( (x + 1) \) from the numerator and denominator:
\[ \lim_{x \to -1} \frac{(x - 3)(x + 1)}{x + 1} \]
Now, simplify further:
\[ \lim_{x \to -1} (x - 3) \]
Now, substitute \( x = -1 \):
\[ (-1 - 3) = -4 \]
So, $\lim\limits_{x \to -1}\frac{x^3 - x^2 - 5x - 3}{(x + 1)^2}=-4$.
