Why does Gauss's law state that there are some 'Finite' number of electric 'Field line' coming out from a charge instead of "Infinite" number? Why does number of field line be a 'finite ' coming out from a charge $Q$ (that is $Q/\epsilon$) where there are 'infinite' number of point around the charge $Q$ each which has a specific value of Electric field strength.
 A: Let me recapitulate the concept.
The field lines are continuous to be drawn in such a way that,


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*the tangent drawn at any point will be parallel to the field at that point.

*the density field lines in a region (NOT at a point) is proportional to the strength of field in that region.
Clearly these statements do not mention anything about the field strength at a point. So let us note couple of points here. Also more importantly, THEY ARE NOT REAL. Yes this is just a visualization tool created by Faraday. 


*

*Field lines concept is a qualitative one. So total number of field lines is not that important. The important point is the field line density. We fix the number of field lines while drawing them. Because we can not draw infinite number of field lines.

*A more quantitative idea would be solid angle. The total solid angle is always $4\pi$ so people generally confusingly use the language that number of field lines is $4\pi$. But they are not. The number of field lines is what you want them to be. 
At last let us see what is the exact statement of Gauss Law.

The net electric flux through any closed surface is equal to $\frac{1}{\varepsilon_o}$ times the net electric charge within that closed surface.

The definition of electric flux is,
$\Phi_E = \iint_S \vec{E}\cdot d\vec{A} \propto$ number of field lines crossing a particular surface S but NOT equal to.
A: Gauss's Law doesn't state that there are a finite number of field lines. Gauss's Law doesn't really tell anything about (at least directly) the number of field lines rather it tells about the flux of the electric field. To make a concrete meaning out of this apparent terminological web, let's define and/or state everything step-by-step. 
Electric Field: It's the vector field whose value at a given point is given by the force acting on a static point-charge of unit charge by virtue of its charge. 
Flux: $\phi = \displaystyle\int_{S}\vec{E}.d\vec{A}$ 
Gauss's Law: $\displaystyle\int_{S}\vec{E}.d\vec{A} = \dfrac{Q_{enc}}{\epsilon_0}$
Flux is not really defined as the number of field lines crossing a particular patch of the surface rather it is just the surface integration of  the scalar product of electric field and elemental surface area. And Gauss's law simply talks about this flux. 
However, when we try to form a picture of the electric field in the terms of things that we are primitively familiar with, we come up with the picture in which some finite number of infinitely long arrows are coming out of a static charge, each pointing radially outward. This model has an inherent deviation from the actual field because the arrows do not occupy the entire space while the field actually does. But we can still go a bit far with this model keeping in mind that it is not really in the best agreement with experiments and should not be given precedence over the mathematical laws derived via experiments while drawing the conclusions. 
Let's denote a quantity defined as the number of arrows crossing a particular patch of the surface by $\phi_p$. Now it is interesting that $\dfrac{\phi_pQ}{N\epsilon_0} = \phi$ where $Q$ is the charge of the static point charge under consideration, $N$ is the number of infinite arrows imagined to be coming out of the considered point charge. So the number of arrows crossing a patch of a surface is in direct proportionality with the electric field flux for a given number of imaginary arrows. But as one can clearly see, this analogy doesn't really tell anything about the number of electric field lines, which if defined under any sensible considerations, would always be either zero or infinite through a finite patch of a surface. 
Therefore, the textbookish claims of the number of field lines coming out of a point charge of charge $q$ being $\dfrac{q}{\epsilon_0}$ are false.$\dfrac{q}{\epsilon_0}$ is just the flux. 
A: Where did you read this? Field lines are just a qualitative measure, you may draw them as much as you like and they may be inifnite in number.
