\lim _{x\to \:0}\left(\frac{x^2-a^2}{x+a}\right)

Given that  a is an unknown real number, find each of the following limits. If needed, use oo for , -oo for , or DNE for "does not exist."

To find the limit

\[
\lim _{x\to 0}\left(\frac{x^2-a^2}{x+a}\right),
\]

let’s approach the problem step by step:

1. **Substitute \( x = 0 \) directly into the expression:**

   \[
   \frac{0^2 - a^2}{0 + a} = \frac{-a^2}{a} = -a.
   \]

Thus, the limit is:

\[
\boxed{-a}.
\]

This is the solution for any real number \( a \), except for \( a = 0 \), which we will handle separately.

2. **Special case \( a = 0 \):**

   Substituting \( a = 0 \) into the original expression:

   \[
   \lim_{x \to 0} \left( \frac{x^2 - 0^2}{x + 0} \right) = \lim_{x \to 0} \frac{x^2}{x} = \lim_{x \to 0} x = 0.
   \]

So, for \( a = 0 \), the limit is:

\[
\boxed{0}.
\]

### Final Answer:
- For \( a \neq 0 \), the limit is \( \boxed{-a} \).
- For \( a = 0 \), the limit is \( \boxed{0} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What happens if we apply L'Hopital's Rule in this problem?
2. How would this limit change if \( a \) was restricted to positive values?
3. Can we generalize this type of limit for other forms of rational functions?
4. What if the denominator was \( x - a \) instead of \( x + a \)?
5. How can we graphically interpret the result of this limit?

**Tip:** When approaching limits, always check if direct substitution is possible before considering alternative methods like factoring or L'Hopital's Rule.

Given that a is an unknown real number, find the limit: \lim _{x\to 0}\left(\frac{x^2-a^2}{x+a}\right).

Explore the step-by-step solution to finding the limit \lim _{x\to 0}\left(\frac{x^2-a^2}{x+a}\right), where a is an unknown real number. This problem covers the application of direct substitution, rational function limits, and the special case when a = 0. Suitable for advanced high school or undergraduate students.

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