To calculate the limit and verify using geometry, let’s analyze the problem step-by-step.

Function:

$f(x) = \frac{15}{2}x + 4$ The interval is $[0,4]$. We are tasked with finding:

$\lim_{N \to \infty} L_N$ where $L_N$ represents the Riemann sum approximation of the definite integral over $[0,4]$.

Step 1: Calculate the exact integral of $f(x)$ over $[0,4]$

The definite integral of $f(x)$ from 0 to 4 is:

$\int_0^4 f(x) \, dx = \int_0^4 \left(\frac{15}{2}x + 4\right) \, dx$

Breaking this into two parts:

$\int_0^4 \frac{15}{2}x \, dx + \int_0^4 4 \, dx$

First term:

$\int_0^4 \frac{15}{2}x \, dx = \frac{15}{2} \int_0^4 x \, dx = \frac{15}{2} \cdot \left[\frac{x^2}{2}\right]_0^4 = \frac{15}{2} \cdot \frac{16}{2} = \frac{15}{2} \cdot 8 = 60$

Second term:

$\int_0^4 4 \, dx = 4 \cdot \int_0^4 1 \, dx = 4 \cdot \left[x\right]_0^4 = 4 \cdot (4 - 0) = 16$

Total integral:

$\int_0^4 f(x) \, dx = 60 + 16 = 76$

Thus, the exact value of $\lim_{N \to \infty} L_N$ is: $\boxed{76}$

Step 2: Verify using geometry

The graph of $f(x)$ is a straight line, so the area under the curve from $x = 0$ to $x = 4$ can be verified geometrically.

1. Trapezoid Approach: Divide the area into a trapezoid with parallel sides $f(0)$ and $f(4)$, and height 4.
   • $f(0) = \frac{15}{2}(0) + 4 = 4$
   • $f(4) = \frac{15}{2}(4) + 4 = 30 + 4 = 34$
   • Trapezoid area: $\text{Area} = \frac{1}{2} \cdot \text{height} \cdot (\text{base}_1 + \text{base}_2)$ $\text{Area} = \frac{1}{2} \cdot 4 \cdot (4 + 34) = 2 \cdot 38 = 76$

Thus, the geometric verification confirms the result: $\boxed{76}$