Combine 2 arrays using recursion

Your stopping condition is faulty.

First of all you compare a certain value of array a and b at position i/j with the length of array a and b, you should check for the index itself.

Secondly, there is only one final step where you should stop - which is both arrays a and b were fully processed (you are currently checking with or).

Based on your current implementation your method should look something like this:

    public int[] combineArray(int[] a, int[] b, int[] c, int i, int j, int x){
        if (i >= a.length && j >= b.length) {
            // only if both arrays are processed stop and return c
            return c;
        } else if (i >= a.length) {
            // we finished with array A, process array B
            c[x] = b[j];
            combineArray(a, b, c, i, j + 1, x + 1);
        } else if (j >= b.length) {
            // we finished with array B, process array A
            c[x] = a[i];
            combineArray(a, b, c, i + 1, j, x + 1);
        } else {
            // both arrays still have elements
            if (a[i] < b[j]) {
                c[x] = a[i];
                combineArray(a, b, c, i + 1, j, x + 1);
            } else {
                c[x] = b[j];
                combineArray(a, b, c, i, j + 1, x + 1);
            }
        }
        return c;
    }

Last but not least, when you print the array you would want to iterate over the array c itself, not call again the find.combineArray method, something like

        for(int i = 0; i < x.length; i++){
            System.out.println("On position : " + i + " is value " + x[i]);
        }

Important: There was a duplicated 8, one 8 in array a and one 8 in array b, yet in the array displaying the results array, there are no two 8´s. Also you did not mention how to handle that, so I think this might be a typo, because the sum of elements of array a and b is one less than the size of the result array c. Thats why I removed one 8. But it would also work with both 8´s.

Try this:

    public void combineArray(int[] a, int[] b, int[] c, int i, int j, int x) {
        if (i == a.length && j == b.length) {
            return;
        }

        if (i == a.length) {
            c[x] = b[j];
            combineArray(a, b, c, i, j + 1, x + 1);
            return;
        }

        if (j == b.length) {
            c[x] = a[i];
            combineArray(a, b, c, i + 1, j, x + 1);
            return;
        }

        if(a[i] <= b[j]) {
            c[x] = a[i];
            combineArray(a, b, c, i + 1, j, x + 1);
            return;
        }

        if(a[i] > b[j]) {
            c[x] = b[j];
            combineArray(a, b, c, i, j + 1, x + 1);
            return;
        }
    }
    
    public static void main(String [] args) {
        FindMe find = new FindMe();
        int[] a = {1, 2, 3, 5, 13, 21, 34, 55, 89};
        int[] b = {4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048};
        int[] c = new int[a.length + b.length];
        find.combineArray(a, b, c, 0, 0, 0);
        for (int i = 0; i < c.length; i++) {
            System.out.println("On position : " + i + " is value " + c[i]);
        }
    }

Here are some points to mention:

1. You are calling the method combineArray() every iteration int the for-loop. Don't do that, just take the length of the array with the results.
2. you dont need that x array, you are already writing all the results into the c array.
3. Therefor you also dont need to return some form of array from the method.
4. Don´t just a literal to instanciate the c array. The length of the c array is the sum of the lengths of the a and b arrays, which you want to combine.
5. Make sure to check the indexes i, j and x before using them for retrieving values from the arrays. Otherwise you might run into OutOfIndexExceptions.
6. Be carefully with formatting your code and values.