Step 1: Understanding the chemical linkage.
The \(-\text{CO}-\text{NH}-\) linkage, also known as an amide group, is found in compounds that are part of the nylon family. These compounds are often used in polymer chemistry.
Step 2: Analyzing the options.
(A) Vinylcyanide: This is a nitrile compound and does not contain the \(-\text{CO}-\text{NH}-\) linkage.
(B) Dimethylterephthalate: This compound is an ester and does not contain an amide group.
(C) \(\varepsilon\)-Caprolactam: Correct — \(\varepsilon\)-Caprolactam contains the \(-\text{CO}-\text{NH}-\) linkage and is the precursor for nylon-6 production.
(D) Hexamethylenediamine: This is a diamine and does not contain an amide group.
Step 3: Conclusion.
The correct answer is (C) \(\varepsilon\)-Caprolactam, as it contains the \(-\text{CO}-\text{NH}-\) linkage.