Hun er lige ved at tabe pagajen, fordi hendes hænder er våde.

It is an idiomatic expression meaning just about to, on the verge of, or often simply almost.

So:

• Hun er lige ved at tabe pagajen = She is just about to drop the paddle / She almost drops the paddle.

A useful thing to remember is that lige ved at is best learned as a whole chunk.

Yes, but the nuance changes a little.

• Hun er ved at tabe pagajen = She is dropping / is about to drop the paddle.
• Hun er lige ved at tabe pagajen = She is very close to dropping it; it emphasizes the near-miss.

So lige adds the idea of just barely / almost.

Because tabe is in the infinitive form, and Danish normally uses at before an infinitive in this kind of structure.

• at tabe = to drop
• at spise = to eat
• at gå = to walk

After lige ved, Danish uses at + infinitive:

• lige ved at falde
• lige ved at grine
• lige ved at tabe pagajen

Because after at, Danish uses the infinitive, not the present tense.

• tabe = to drop
• taber = drops / is dropping

So:

• Hun taber pagajen = She drops / is dropping the paddle.
• Hun er lige ved at tabe pagajen = She is just about to drop the paddle.

By itself, ved often does mean by, at, or near. But here it is part of a fixed expression:

• være ved at + infinitive
• være lige ved at + infinitive

In this expression, you should not translate ved word by word. Learn the whole phrase as one unit.

Because pagajen is the definite form, meaning the paddle.

Danish often adds the definite article to the end of the noun:

• en pagaj = a paddle
• pagajen = the paddle

The sentence refers to a specific paddle, probably the one she is holding, so the definite form is natural.

Danish often uses the definite form where English might use a possessive, especially when the thing is obvious from the situation.

So tabe pagajen can sound perfectly natural even if English might say drop her paddle.

If you want to make the ownership explicit, Danish would normally use the reflexive possessive here:

• Hun er lige ved at tabe sin pagaj = She is just about to drop her own paddle.

Using hendes pagaj in a sentence with hun would usually suggest another female person’s paddle, not her own.

This is a very common point of confusion.

Sin/sit/sine normally refers back to the subject of the same clause. But in:

• fordi hendes hænder er våde

the possessive is inside the subject phrase of the fordi-clause. In that position, Danish uses hendes, not sine.

Compare:

• Hun vasker sine hænder. = She washes her own hands.
  Here hun is the subject of the same clause, so sine works.

• ... fordi hendes hænder er våde. = ... because her hands are wet.
  Here hendes is the natural form.

So even though the hands belong to her, sine is not used in this sentence structure.

Because adjectives in Danish agree with number and, in some cases, gender.

The basic forms are:

• våd for common gender singular
• vådt for neuter singular
• våde for plural

So:

• en hånd er våd = a hand is wet
• et gulv er vådt = a floor is wet
• hænder er våde = hands are wet

Since hænder is plural, the adjective must be våde.

Fordi means because and introduces a subordinate clause.

In Danish, subordinate clauses do not use normal main-clause verb-second word order. The usual pattern is:

• fordi + subject + finite verb + ...

So:

• fordi hendes hænder er våde

That is normal subordinate-clause order.

A useful extra example:

• Hun taber næsten pagajen, fordi hendes hænder ikke er tørre.

Notice that in the subordinate clause, ikke comes before the finite verb’s position would be in a main clause:

• fordi hendes hænder ikke er tørre

It can mean both, depending on context.

• tabe nøglerne = lose the keys
• tabe glasset = drop the glass

In this sentence, with pagajen, the meaning is clearly drop.