Evaluate the limit: $$ \lim _{x \rightarrow-1} \frac{x^{2}-x-2}{x+1} $$ Question Help: Video Message instructor Submit Question Jump to Answer Sum

Question

UpStudy AI · Accepted Answer

To evaluate the limit:

$$
\lim_{x \rightarrow -1} \frac{x^{2} - x - 2}{x + 1}
$$

**Step 1: Substitute $$ x = -1 $$ directly into the expression to check for indeterminate forms.**

$$
\frac{(-1)^2 - (-1) - 2}{-1 + 1} = \frac{1 + 1 - 2}{0} = \frac{0}{0}
$$

Since we obtain an indeterminate form $$ \frac{0}{0} $$, we can apply algebraic techniques to simplify the expression.

**Step 2: Factor the numerator.**

The numerator is a quadratic expression:

$$
x^2 - x - 2
$$

We look for two numbers that multiply to $$-2$$ and add to $$-1$$. These numbers are $$-2$$ and $$1$$.

$$
x^2 - x - 2 = (x - 2)(x + 1)
$$

**Step 3: Simplify the expression by canceling common factors.**

$$
\frac{(x - 2)(x + 1)}{x + 1}
$$

For $$ x \neq -1 $$, we can cancel $$ (x + 1) $$ from the numerator and the denominator:

$$
\frac{(x - 2)\cancel{(x + 1)}}{\cancel{x + 1}} = x - 2
$$

**Step 4: Evaluate the simplified expression at $$ x = -1 $$.**

$$
\lim_{x \rightarrow -1} (x - 2) = (-1) - 2 = -3
$$

**Conclusion:**

The limit is **$$-3$$**.
The limit is $$-3$$.

Evaluate the limit: \( \lim _{x \rightarrow-1} \frac{x^{2}-x-2}{x+1} \) Question Help: Video Message instructor Submit Question Jump to Answer Sum