A tree has 10 pounds of apples at dawn.  Every afternoon, a bird comes and eats x pounds of apples.  Overnight, the amount of food on the tree increases by 10%.  What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
1. Let \( A_n \) be the amount of apples on the tree at the end of day \( n \). Initially, \( A_0 = 10 \) pounds.
2. Each afternoon, the bird eats \( x \) pounds of apples, so the amount of apples left after the bird eats is \( A_n - x \).
3. Overnight, the amount of apples increases by 10%, so the amount of apples at the start of the next day is \( 1.1 \times (A_n - x) \).

To sustain the bird indefinitely, the amount of apples at the start of each day must be the same, i.e., \( A_{n+1} = A_n \). Therefore, we set up the equation:
\[ A_{n+1} = 1.1 \times (A_n - x) \]
Since \( A_{n+1} = A_n \), we have:
\[ A_n = 1.1 \times (A_n - x) \]

4. Substitute \( A_n = 10 \) pounds (the initial amount of apples):
\[ 10 = 1.1 \times (10 - x) \]

5. Solve for \( x \):
\[ 10 = 1.1 \times (10 - x) \]
\[ 10 = 11 - 1.1x \]
\[ 10 - 11 = -1.1x \]
\[ -1 = -1.1x \]
\[ x = \frac{1}{1.1} \]
\[ x = \frac{10}{11} \]

Therefore, the maximum value of \( x \) such that the bird can sustain itself indefinitely on the tree without the tree running out of food is \( \boxed{\frac{10}{11}} \).