The user wants me to solve a problem involving a functional equation with an integral.

1. Understand the given functional equation: We are given a continuous function $f: R \to R$ that satisfies the equation: $f(x) = \int\limits_0^x {f(t)dt}$
This equation must hold for all $x \in R$.

2. Find an initial condition: Let's evaluate the equation at $x=0$. $f(0) = \int\limits_0^0 {f(t)dt}$ The definite integral from a point to itself is always zero. Therefore, $f(0) = 0$ This gives us an initial condition for the function $f(x)$.

3. Form a differential equation: Since $f(x)$ is given to be continuous, we can differentiate both sides of the given functional equation with respect to $x$. We will use the Leibniz rule (or the first Fundamental Theorem of Calculus) for differentiating the integral part.

According to the Fundamental Theorem of Calculus, if $F(x) = \int_a^x g(t)dt$, then $F'(x) = g(x)$ for a continuous function $g$.

Applying this to our equation: $\frac{d}{dx} f(x) = \frac{d}{dx} \left( \int\limits_0^x {f(t)dt} \right)$
$f'(x) = f(x)$
This is a first-order linear ordinary differential equation.

4. Solve the differential equation: We need to solve the differential equation $f'(x) = f(x)$, or $\frac{dy}{dx} = y$ where $y = f(x)$. We can solve this by separating variables: $\frac{dy}{y} = dx$
Integrating both sides: $\int \frac{dy}{y} = \int dx$
$\ln|y| = x + C_1$
where $C_1$ is the constant of integration.

To solve for $y$, we exponentiate both sides: $|y| = e^{x+C_1} = e^x \cdot e^{C_1}$
$y = \pm e^{C_1} e^x$
Let $C = \pm e^{C_1}$. Then the general solution for $f(x)$ is: $f(x) = Ce^x$
where $C$ is an arbitrary constant.

5. Determine the constant C: We use the initial condition we found in step 2, which is $f(0) = 0$. Let's substitute $x=0$ into our general solution: $f(0) = Ce^0$
$0 = C \cdot 1$
$C = 0$

6. Determine the function f(x): Substituting the value of $C=0$ back into the general solution $f(x) = Ce^x$, we get: $f(x) = 0 \cdot e^x = 0$
So, the only continuous function that satisfies the given equation is the zero function, $f(x) = 0$ for all $x \in R$.

7. Calculate the required value: The question asks for the value of $f(\ln 5)$. Since $f(x) = 0$ for all $x$, we have: $f(\ln 5) = 0$

Final Answer: The value of $f(\ln 5)$ is 0.