Why in calculus the angles are measured in radians?

Definition of 1 radian:

Angle subtended at the centre of a circle by an arc equal in length to the radius.

Definition of 1 degree:

The angle subtended by one three-hundred-and-sixtieth of the circumference of a circle.

Definition of 1 gradian:

The gradian is a unit of measurement of an angle, equivalent to $1 \over 400$ of a turn, $9\over 10$ of a degree or $\pi \over 200$ of a radian.

Clearly, the radian has the most concrete definition: a ratio of two distances. It doesn't require any other thing for its support.

In the definition of degrees, you have to know the definition of an angle (to measure circumference) before you can divide it by $360$.

And clearly, gradians are helpless without degrees or radians.

So, isn't the choice obvious?

Look at the proof of the limit you are asking. And no, using L'Hospital is not a proof. You will see that the proof fails if you use degrees instead of radians, and the main point is that you can compare the numerical value of radians with length in a very direct way, just by the definition of radians. So you can say something like $\sin x \leq x$ when $x$ is in radians, but not when it in degrees.