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To demonstrate the limit property as requested, we need to prove that if:

\[
\lim_{x \to a} f(x) = \ell \quad \text{and} \quad \lim_{x \to a} g(x) = t
\]

then:

\[
\lim_{x \to a} (f(x) + g(x)) = \ell + t
\]

### Proof using the definition of a limit:

By the definition of a limit, for any function \( f(x) \), if \( \lim_{x \to a} f(x) = \ell \), it means that for every \( \epsilon > 0 \), there exists a \( \delta_1 > 0 \) such that:

\[
0 < |x - a| < \delta_1 \implies |f(x) - \ell| < \frac{\epsilon}{2}
\]

Similarly, for \( g(x) \), since \( \lim_{x \to a} g(x) = t \), for every \( \epsilon > 0 \), there exists a \( \delta_2 > 0 \) such that:

\[
0 < |x - a| < \delta_2 \implies |g(x) - t| < \frac{\epsilon}{2}
\]

Now, we want to prove that \( \lim_{x \to a} (f(x) + g(x)) = \ell + t \). Using the limit definition, this means for every \( \epsilon > 0 \), we need to find a \( \delta > 0 \) such that:

\[
0 < |x - a| < \delta \implies |(f(x) + g(x)) - (\ell + t)| < \epsilon
\]

We start by manipulating the expression:

\[
|(f(x) + g(x)) - (\ell + t)| = |(f(x) - \ell) + (g(x) - t)|
\]

Using the triangle inequality:

\[
|(f(x) - \ell) + (g(x) - t)| \leq |f(x) - \ell| + |g(x) - t|
\]

Since \( |f(x) - \ell| < \frac{\epsilon}{2} \) and \( |g(x) - t| < \frac{\epsilon}{2} \) whenever \( 0 < |x - a| < \min(\delta_1, \delta_2) \), we have:

\[
|f(x) - \ell| + |g(x) - t| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon
\]

Thus, \( |(f(x) + g(x)) - (\ell + t)| < \epsilon \), which proves that:

\[
\lim_{x \to a} (f(x) + g(x)) = \ell + t
\]

This concludes the proof.

---

Do you have any questions or would you like further details on this proof?

### Related Questions:
1. How can we extend this proof to more than two functions?
2. What is the role of the triangle inequality in limit proofs?
3. How would the proof change if one of the limits was infinite?
4. Can this limit rule be applied to discontinuous functions?
5. How does this proof connect to the concept of continuity?

**Tip:** The triangle inequality is a powerful tool when dealing with limits and absolute values, often simplifying the process of proving inequalities.

Demostrar por definición que si lim (x → a) f(x) = ℓ y lim (x → a) g(x) = t entonces lim (x → a) (f(x) + g(x)) = ℓ + t.

This problem demonstrates how to prove the sum of limits in calculus using the definition of a limit. It involves the concepts of limits, triangle inequality, and continuity, suitable for advanced high school or university-level calculus students.

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