The correct option is 98.

An expression of the form A : B :: C : D represents a proportion, which means that the ratio of the first two terms is equal to the ratio of the last two terms. This relationship can be mathematically written as:

$$\frac{A}{B}=\frac{C}{D}$$

In this problem, the given proportion is 36 : 84 :: 42 : X. By mapping these values to our general proportion formula, we have:
A = 36
B = 84
C = 42
D = X (the unknown value we need to find)

Substituting these values into the ratio equation gives:

$$\frac{36}{84}=\frac{42}{X}$$

To solve for X, we can first simplify the fraction on the left side of the equation. Both 36 and 84 are divisible by their greatest common divisor, which is 12:
36 ÷ 12 = 3
84 ÷ 12 = 7

So, the simplified equation is:

$$\frac{3}{7}=\frac{42}{X}$$

Next, we can solve for X by cross-multiplying:

$$3\cdot X=42\cdot 7$$

Calculating the right side of the equation:
42 × 7 = 294

Now, we divide both sides by 3 to find X:

$$X=\frac{294}{3}$$

$$X=98$$

Therefore, the value of X is 98.