Ils vivent près de la frontière entre deux pays.

Ils is a subject pronoun (like they in English). It must be used before a verb:

• Ils vivent = They live

Eux is a stress (disjunctive) pronoun, used in other positions, for emphasis or after prepositions:

• Eux, ils vivent près de la frontière. = They, they live near the border.
• C’est pour eux. = It’s for them.

So, when “they” is the subject of the verb, you use ils, not eux.

Vivent is:

• Verb: vivre (to live)
• Tense: present indicative
• Person: 3rd person plural (they)
• Form: ils vivent

The full present conjugation of vivre:

• je vis
• tu vis
• il/elle/on vit
• nous vivons
• vous vivez
• ils/elles vivent

The ending -ent marks 3rd person plural in writing, but it is not pronounced (see next question).

Vivent is pronounced roughly like “veev” /viv/.

For regular present-tense verbs in -er, -re, and many -ir verbs, the written endings -ent in the 3rd person plural (ils/elles) are silent:

• ils parlent → [parl]
• ils prennent → [prenn]
• ils vivent → [viv]

So you pronounce ils vivent as [il viv], not [il vivent].

Yes, there can be a liaison.

• In careful or formal speech: ils vivent → [il‿zviv] (you hear a z sound linking them).
• In everyday casual speech, you may sometimes hear it without liaison: [il viv].

The liaison here is optional but common in standard, careful French.

Both can translate as “live”, but they are used differently:

• Habiter = to live in a place (reside)

  • Ils habitent à Paris. = They live in Paris.
  • Ils habitent près de la frontière. is also possible.

• Vivre is broader: to live one’s life / exist / experience life

  • Ils vivent près de la frontière. = They live (their lives) near the border.
  • Il vit heureux. = He lives happily.
  • Je vis avec mes parents. = I live with my parents.

In your sentence, vivent emphasizes their life situation rather than just their address. But habitent près de la frontière would also be correct and natural.

Près almost always needs de when it is followed by a noun:

• près de la frontière = near the border
• près de la gare = near the station
• près de chez moi = near my place

Près alone is usually an adverb, not directly followed by a noun:

• Il est tout près. = He is very near.

So you say près de + [thing].

Because frontière is feminine, so the article is la, not le:

• la frontière = the border

The structure is:

• près de + la frontière → près de la frontière

If the noun were masculine singular, you’d get du (de + le):

• près du lac = near the lake
• près du pont = near the bridge

Frontière is feminine:

• la frontière = the border
• une frontière = a border
• cette frontière = this border

Its gender affects:

• The article: la frontière, not le.
• Adjectives agreeing with it:
  • la frontière nationale (feminine adjective)
  • les frontières internationales (feminine plural)

Both are possible, but they are not identical:

• près de la frontière = near the border (in the area close to it)
• à la frontière = at the border (right there, on the border)

Examples:

• Ils vivent près de la frontière.
  They live in a town/village close to the border region.

• Ils travaillent à la frontière.
  They work at the border itself (e.g. as customs officers).

Yes, proche de la frontière is possible, but with a slightly different feel:

• près de la frontière = near the border (very common, neutral)
• proche de la frontière = close to the border (often a bit more formal or written style)

So:

• Ils vivent près de la frontière.
• Ils vivent proche de la frontière.

Both are acceptable; près de is more frequent in everyday speech.

Both are grammatically correct, but the meaning changes:

• entre deux pays = between two countries (unspecified, any two)
• entre les deux pays = between the two countries (specific, already known from context)

Your sentence:

• Ils vivent près de la frontière entre deux pays.
  → They live near the border between two (unspecified) countries.

If the conversation had already identified which two countries, you could say:

• Ils vivent près de la frontière entre les deux pays.

French, like English, can use a number + noun without an article when it’s indefinite:

• deux pays = two countries (any two)
• trois villes = three cities

You add les when you mean specific ones already known:

• les deux pays = the two countries (those two, which we know)

So:

• entre deux pays = between two countries (in general)
• entre les deux pays = between the two countries (specific pair)

Pays is masculine, and its spelling is the same in singular and plural:

• Singular: un pays = a country
• Plural: des pays = countries

Only the article (and sometimes verbs/adjectives) show the number:

• Ce pays est grand. = This country is big.
• Ces pays sont grands. = These countries are big.