The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.
1. **Define the function and properties:**
   Given the function \( f(x) = \frac{ax + b}{cx + d} \) with properties \( f(19) = 19 \), \( f(97) = 97 \), and \( f(f(x)) = x \) for all \( x \neq -\frac{d}{c} \).

2. **Consider the function \( h(x) = f(x) - x \):**
   Since \( f(19) = 19 \) and \( f(97) = 97 \), the equation \( h(x) = 0 \) has roots at \( x = 19 \) and \( x = 97 \).

3. **Express \( f(x) - x \):**
   \[
   f(x) - x = \frac{ax + b}{cx + d} - x = \frac{ax + b - x(cx + d)}{cx + d} = \frac{ax + b - cx^2 - dx}{cx + d} = \frac{-(cx^2 - (a-d)x + b)}{cx + d}
   \]
   For \( f(x) - x = 0 \), the numerator must be zero:
   \[
   -cx^2 + (a-d)x + b = 0
   \]
   Since the roots are \( x = 19 \) and \( x = 97 \), we can write:
   \[
   -c(x - 19)(x - 97) = -cx^2 + c(19 + 97)x - c \cdot 19 \cdot 97
   \]
   Comparing coefficients, we get:
   \[
   a - d = c(19 + 97) = 116c \quad \text{and} \quad b = -c \cdot 19 \cdot 97
   \]

4. **Substitute values into \( f(x) \):**
   \[
   f(x) = \frac{ax + b}{cx + d} = \frac{(d + 116c)x - c \cdot 19 \cdot 97}{cx + d}
   \]

5. **Use the condition \( f(f(x)) = x \):**
   \[
   f\left( \frac{(d + 116c)x - c \cdot 19 \cdot 97}{cx + d} \right) = x
   \]
   This implies that the function \( f(x) \) must be its own inverse.

6. **Determine specific values for \( a \), \( b \), \( c \), and \( d \):**
   From the given conditions, we can solve for \( c \) and \( d \):
   \[
   c = -1, \quad b = 19 \cdot 97, \quad a - d = 116
   \]
   Substituting \( c = -1 \):
   \[
   b = 19 \cdot 97 = 1843
   \]
   \[
   a - d = 116
   \]
   Using the condition \( f(f(x)) = x \), we find:
   \[
   f(0) = \frac{19 \cdot 97}{d} = \frac{1843}{d}
   \]
   \[
   f\left( \frac{1843}{d} \right) = 0
   \]
   Solving for \( d \):
   \[
   1843 - \frac{116 \cdot 1843}{d} + 1843 = 0 \implies d = 58
   \]
   \[
   a = -58
   \]

7. **Final form of \( f(x) \):**
   \[
   f(x) = \frac{-58x + 1843}{-x + 58}
   \]

8. **Determine the unique number not in the range of \( f \):**
   The value not in the range of \( f(x) \) is the limit as \( x \to \infty \):
   \[
   \lim_{x \to \infty} f(x) = \frac{-58}{-1} = 58
   \]

The final answer is \( \boxed{58} \).