This integral does not converge as \(x \to \infty\)

Calculus Level 5

For \(x > 0\), define \(f(x)\) as follows: \[f(x) = \int_{0}^{x} \left( \sqrt{t^2+1} - t \right) \, dt.\] Let \(L = \displaystyle\lim_{x\to\infty} \left(f(x) - \dfrac{\ln x}{2}\right).\)

If \(L = \dfrac{a + \ln{b}}{c}\) for positive integers \( a, b,\) and \(c\) with \( \gcd(a,c) = 1 \), then find the value of \( a + b + c \).