Could anyone please suggest on how to be confortable with using infinitesimals via any analogy or something?Infinitesimals are widely used in physics,engineering and other areas of science instead of relying on limits and they are easier to use.But while applying them,the "mathy" side of mine whispers that "this is not a perfect calculation,only an approximation".I want to get rid of this feeling and want to use infinitesimals with more comfort.Thanks in advance.
As @joe mentions in a comment, Jerome Keisler's (free, on-line) calculus book develops the subject from a rigorous viewpoint using infinitesimals.
Edward Nelson's reformulation (of A. Robinson's original) is also more user-friendly. This nicely-packaged version is discussed in a wonderful little book "Non-standard Analysis" by Alain Robert. Robert's book does not discuss the model-theory background, but, rather, primarily how to correctly use the ideas. Even if you want to look at proofs later, Robert's description of the function/use of the ideas is very illuminating.