limit of f(x) as x tends to a is actually taken when x is not equal to a but very close to a.
let me explain more.
suppose you buy a shirt costing Rs. 500.
the shop keepertells you that if you dont have money right noe, it's ok
you keep paying half the balance, every week, when you finish, the shirt becomes yours.
now ist week you will pay 250
next week 125
now you will see that half of the balance becomes very small as the weeks pass y but the balance will never ecome exactly zero, it'll keep ' tending' to zero, going very close to zero.
this is the condition that balance tnds to zero.
hope its clear now.
derivative is rate measure.also geometrically it's the slope of a tangent to a curve at a given point.
inegration is actually the area under a curve. it's found by dividing the curve into small, narrow strips and adding areas of all of them.
hope it's becoming more clear to you.
do write back for further clarifiaction.
its good that you want to get the concepts clear rather than just doing the questions mechanically.
all the best...