1.1 Limit, Continuity and Differentiability (BT103HS)

A limit is the value that a function "approaches" as the input approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, and integrals.

We write: limx→a f(x) = L

A function f(x) is continuous at a point x = a if the following three conditions are met:

1. f(a) is defined.
2. limx→a f(x) exists.
3. limx→a f(x) = f(a).

A function is differentiable at x = a if its derivative exists at that point. Geometrically, this means the graph of the function has a non-vertical tangent line at (a, f(a)).

All differentiable functions are continuous, but not all continuous functions are differentiable (e.g., f(x) = |x| at x = 0).