“I can definitely tell you're getting better.”

Your friend is learning to play the guitar. He's frustrated because he doesn't think he's improving. But you do think he's improving. You say:

I can definitely tell you're getting better.

(someone) can tell (clause)

When you "can tell" something, it means that you know it by experiencing it:

I can tell that you're lying to me.

Could you tell I wanted to leave?

You can leave "that" out in casual conversation, but in formal speaking or in writing you should definitely include it:

I can definitely tell that you're improving.

definitely (do something)

When you use "definitely", it means that you're sure.

"Definitely" is often used together with "can tell":

I can definitely tell that you've lost some weight.

(be) getting better

When you're "getting better", it means you're improving. "Get better" is a more casual version of "improve".

"Get better" also means to recover after a sickness or injury, by the way.

“Hold on — you might be able to recover it.”

Memorize

Your wife was working on a document on the computer at home. The program crashed, and she's upset because she thinks all of the work she did is lost. You think she might be able to get the file back again. You say:

Hold on — you might be able to recover it.

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