Simplify the complex fraction.
\[
\frac{\frac{x-3}{x^{2}-16}}{1+\frac{1}{x-4}}
\]

Step 1 ：Given the complex fraction \(\frac{\frac{x-3}{x^{2}-16}}{1+\frac{1}{x-4}}\)

Step 2 ：First, simplify the denominator. The denominator is a sum of 1 and a fraction. We can simplify this by finding a common denominator and adding the fractions. The common denominator is (x-4).

Step 3 ：So, the denominator simplifies to \(1+\frac{1}{x-4} = \frac{x-4+1}{x-4} = \frac{x-3}{x-4}\)

Step 4 ：Now, simplify the entire fraction by multiplying the numerator and the denominator by the reciprocal of the denominator.

Step 5 ：So, the complex fraction simplifies to \(\frac{x - 3}{(x - 3)(x + 4)}\)

Step 6 ：Finally, simplify the fraction to get \(\frac{1}{x+4}\)

Step 7 ：Final Answer: The simplified form of the complex fraction is \(\boxed{\frac{1}{x+4}}\)