What is the apparent magnitude of the weakest stars we can see in one telescope on the ground with a diameter of 2.4m? All I know is its main mirror diameter; 2.40 m

I also know that the eye, which its pupil is 5mm in diameter can see up to an apparent magnitude of 6.

How can I calculate the apparent magnitude that the telescope can see?

I have used the equation of 

$5⋅log(\frac{D_1}{D_0})$
However, it gave the answer of 13.40, while the answer should be 19.40
You just need to add the 13.4  that you got to the 6 that you can see without a telescope and that's it.  The formula is telling you the difference in magnitude that arises from the different apertures of the eye and the telescope.