I dag lærer vi at bruge "for ... til at" og "nok til at" i danske sætninger.

Because Danish main clauses usually follow the verb-second rule (V2). The finite verb must come in the second position.

So when I dag comes first, the verb lærer must come next, and the subject vi comes after it:

• I dag lærer vi ...

If the subject comes first, then you get:

• Vi lærer i dag ...

Both are correct, but the emphasis is slightly different. Starting with I dag puts focus on today.

In standard Danish, i dag is normally written as two words. It means today.

So:

• i dag = correct modern spelling

You may sometimes see idag in informal writing, but i dag is the standard form learners should use.

Because bruge is in the infinitive form, and Danish usually uses at before an infinitive, much like English uses to.

So:

• at bruge = to use

In the sentence:

• lærer vi at bruge ...

this works like English:

• we are learning to use ...

They do not all do exactly the same job, even though they look the same.

1. In at bruge, at marks the infinitive:

   • at bruge = to use

2. In til at, the word til is part of a fixed pattern, and at again introduces an infinitive:

   • til at + infinitive

So in:

• nok til at forstå
• for træt til at arbejde

the at is still the infinitive marker, but it comes after til because that is how the pattern is built.

The dots show that something goes in the middle. Usually that is an adjective or adverb phrase.

The pattern is:

• for + adjective/adverb + til at + infinitive

It usually means too ... to ...

Examples:

• for ung til at stemme = too young to vote
• for dyr til at købe = too expensive to buy
• for sent til at ringe = too late to call

So the dots are just a teaching shortcut showing there is a slot to fill.

No. This is a very important difference.

• for at usually means in order to
• for ... til at usually means too ... to

Compare:

• Jeg læser dansk for at lære mere.
  = I study Danish in order to learn more.

• Jeg er for træt til at læse dansk.
  = I am too tired to study Danish.

So even though both contain for and at, they are different structures with different meanings.

Nok til at usually means enough to.

A very common pattern is:

• adjective + nok til at + infinitive

Examples:

• gammel nok til at køre = old enough to drive
• stærk nok til at løfte den = strong enough to lift it
• modig nok til at spørge = brave enough to ask

You can also use it with nouns:

• penge nok til at købe det = enough money to buy it

So nok til at expresses that the amount or degree is sufficient.

Because sætninger is plural, and the adjective must agree with it.

• en dansk sætning = a Danish sentence
• danske sætninger = Danish sentences

In plural, the adjective usually takes -e:

• dansk → danske

That is why the sentence says:

• i danske sætninger

Yes, absolutely.

Both are grammatical:

• I dag lærer vi at bruge ...
• Vi lærer i dag at bruge ...

The difference is mostly focus:

• I dag lærer vi ... emphasizes today
• Vi lærer i dag ... starts more neutrally with we

This is a normal consequence of Danish word order and the V2 rule.

Not always, but they are very natural here.

They are used because the sentence is talking about the expressions themselves, not using them in a normal sentence. In English, you would often do the same in a grammar explanation.

So the quotation marks help show:

• these are language forms being discussed
• they are examples of grammar patterns

In ordinary running text, quotation marks would not always be needed, but in teaching material they are very common.