Find the limit as X approaches -1 in X to power 4 + 3 X power 3 - X ^2 + X + 4 all over X + 1

To find the limit as \( x \) approaches \(-1\) of the expression 

\[
\frac{x^4 + 3x^3 - x^2 + x + 4}{x + 1},
\]

we can start by directly substituting \( x = -1 \) into the expression to check if it results in an indeterminate form like \( \frac{0}{0} \). If it does, we'll need to use algebraic techniques like factorization or L'Hôpital's rule.

### Step 1: Substitute \( x = -1 \)

Substitute \( x = -1 \) into the numerator and the denominator:

**Numerator:**
\[
(-1)^4 + 3(-1)^3 - (-1)^2 + (-1) + 4
= 1 - 3 - 1 - 1 + 4 = 0
\]

**Denominator:**
\[
(-1) + 1 = 0
\]

So, the expression results in \( \frac{0}{0} \), which is an indeterminate form. We need to proceed with further simplification.

### Step 2: Factor the numerator

Since \( x = -1 \) is a root of the polynomial \( x^4 + 3x^3 - x^2 + x + 4 \), the numerator is divisible by \( x + 1 \). Let's perform polynomial division or factorization:

\[
x^4 + 3x^3 - x^2 + x + 4 = (x + 1)(x^3 + 2x^2 - 3x + 4)
\]

So, the expression becomes:

\[
\frac{(x + 1)(x^3 + 2x^2 - 3x + 4)}{x + 1}
\]

Since \( x + 1 \) is in both the numerator and the denominator, they cancel out (except when \( x = -1 \)):

\[
\lim_{x \to -1} \frac{x^4 + 3x^3 - x^2 + x + 4}{x + 1} = \lim_{x \to -1} (x^3 + 2x^2 - 3x + 4)
\]

### Step 3: Substitute \( x = -1 \) into the simplified expression

\[
(-1)^3 + 2(-1)^2 - 3(-1) + 4 = -1 + 2 + 3 + 4 = 8
\]

### Conclusion:
The limit as \( x \) approaches \(-1\) of the given expression is \( \boxed{8} \).

Would you like a detailed breakdown of any steps, or do you have any further questions?

Here are some related questions for further practice:

1. What is the limit as \( x \) approaches 2 for the expression \( \frac{x^3 - 8}{x - 2} \)?
2. How would you find the limit as \( x \) approaches 1 of \( \frac{x^3 - 1}{x - 1} \)?
3. Determine the limit as \( x \) approaches 0 for \( \frac{\sin(x)}{x} \).
4. Evaluate the limit as \( x \) approaches infinity of \( \frac{2x^2 + 3x + 1}{x^2 - 5x + 6} \).
5. Find the limit as \( x \) approaches 0 of \( \frac{e^x - 1}{x} \).

**Tip:** When you get an indeterminate form like \( \frac{0}{0} \), it usually indicates that factorization, simplification, or applying L'Hôpital's rule might be necessary to resolve the limit.

Find the limit as X approaches -1 in X to power 4 + 3 X power 3 - X ^2 + X + 4 all over X + 1.

Learn how to find the limit as x approaches -1 for the expression x^4 + 3x^3 - x^2 + x + 4 over x + 1. This solution involves key calculus concepts such as limits, polynomial division, and the application of the Factor Theorem. Suitable for students in Grades 11-12.

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