When we first study limits in introductory calculus we get the intuition that $$\lim\limits_{x\rightarrow c} f(x)= L$$ means as x approaches c, the values of f(x) approaches L.
However in real analysis course, while learning the $\epsilon$-$\delta$ definition of limits, we express the meaning of the above expression as,
for values of f(x) in a certain neighbourhood of L, the preimages of f(x) i.e., x lies in a certain neighbourhood of c.
Which in other words means as f(x) approaches L the preimage approaches c. So why is this definition counterintuitive? Are there any other alternative definitions?