find the limit as x approaches -1 of x+1/1-x^2

We are tasked with finding the limit of the function \( \frac{x + 1}{1 - x^2} \) as \( x \) approaches \(-1\).

### Step 1: Write the expression
The given expression is:
\[
\lim_{x \to -1} \frac{x + 1}{1 - x^2}
\]

### Step 2: Simplify the denominator
Notice that \( 1 - x^2 \) is a difference of squares:
\[
1 - x^2 = (1 - x)(1 + x)
\]

Thus, the expression becomes:
\[
\lim_{x \to -1} \frac{x + 1}{(1 - x)(1 + x)}
\]

### Step 3: Cancel common factors
We can cancel out the common factor of \( (x + 1) \) in the numerator and denominator:
\[
\lim_{x \to -1} \frac{1}{1 - x}
\]

### Step 4: Substitute \( x = -1 \) into the simplified expression
Now, substitute \( x = -1 \) into the simplified expression:
\[
\frac{1}{1 - (-1)} = \frac{1}{1 + 1} = \frac{1}{2}
\]

### Conclusion:
The limit is:
\[
\boxed{\frac{1}{2}}
\]

Would you like more details, or do you have any further questions?

Here are 5 related questions for practice:
1. Find the limit as \( x \to 1 \) of \( \frac{x - 1}{1 - x^2} \).
2. Evaluate the limit \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \).
3. Determine the limit \( \lim_{x \to 0} \frac{x^2 + 2x}{x} \).
4. Compute \( \lim_{x \to -1} \frac{x^2 - 1}{x + 1} \).
5. Find the limit \( \lim_{x \to 3} \frac{x^2 - 9}{x - 

Learn how to find the limit as x approaches -1 for the function (x + 1) / (1 - x^2) with a clear step-by-step solution. This problem explores key algebraic concepts such as limits and the difference of squares. Ideal for high school students in Grades 11-12.

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