To use the $\epsilon-\delta$ definition of a limit to establish that:

$\lim_{x \to 1} \frac{x}{1+x} = \frac{1}{2},$

we proceed as follows:

Step 1: Recall the $\epsilon-\delta$ definition of a limit

We want to show that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - 1| < \delta$, then:

$\left| \frac{x}{1+x} - \frac{1}{2} \right| < \epsilon.$

Step 2: Manipulate the expression

We aim to find how $\left| \frac{x}{1+x} - \frac{1}{2} \right|$ depends on $x$, and then determine $\delta$ in terms of $\epsilon$.

First, simplify $\left| \frac{x}{1+x} - \frac{1}{2} \right|$:

$\left| \frac{x}{1+x} - \frac{1}{2} \right| = \left| \frac{2x - (1+x)}{2(1+x)} \right| = \left| \frac{x-1}{2(1+x)} \right|.$

Thus, we need to ensure that:

$\left| \frac{x-1}{2(1+x)} \right| < \epsilon.$

Step 3: Bound the denominator

We need to find a bound for $\frac{1}{1+x}$ around $x = 1$. If $x$ is close to 1, say $|x-1| < 1$, then $0 < x < 2$, which implies that:

$1 < 1+x < 3.$

Therefore, $\frac{1}{1+x}$ is bounded between $\frac{1}{2}$ and 1. In particular, we can approximate $\frac{1}{1+x} \leq 1$ when $x$ is sufficiently close to 1. Thus:

$\left| \frac{x-1}{2(1+x)} \right| \leq \frac{|x-1|}{2}.$

Step 4: Choose $\delta$

To satisfy $\left| \frac{x-1}{2(1+x)} \right| < \epsilon$, we require that:

$\frac{|x-1|}{2} < \epsilon.$

Multiplying both sides by 2 gives:

$|x - 1| < 2\epsilon.$

Therefore, we can choose $\delta = 2\epsilon$.

Step 5: Conclusion

By choosing $\delta = 2\epsilon$, we ensure that for every $\epsilon > 0$, if $|x - 1| < \delta$, then:

$\left| \frac{x}{1+x} - \frac{1}{2} \right| < \epsilon.$

This completes the proof using the $\epsilon-\delta$ definition of a limit.

Do you want further clarification on any part of the process?

Here are some related questions:

1. How can the $\epsilon-\delta$ proof be generalized for different limits?
2. Can you apply the same approach to show limits at infinity?
3. What are some examples of discontinuous functions and their limits?
4. How does the choice of $\delta$ affect the precision of the limit proof?
5. What are common pitfalls in applying the $\epsilon-\delta$ definition?

Tip: The key to mastering the $\epsilon-\delta$ proof technique is practice with different types of functions, especially more complex rational and trigonometric functions.