Basic Limits & Continuity (DP IB Analysis & Approaches (AA))

A limit in mathematics is the tendency of a mathematical process as it approaches, but never quite reaches, an 'end point' of some sort.

E.g. the limit of as gets bigger and bigger is zero, because tends to zero as tends to infinity.

The notation denotes "the limit of the function as goes to (or approaches) a".

False.

The limit is concerned with the function's behaviour as it approaches the point, not its value at the point.

represents the limit of the function as gets infinitely big in the positive direction

A continuous function is a function whose graph can be drawn without lifting the pencil from the paper, with no holes or sudden jumps.

A function is differentiable at a point if its derivative exists and has a well-defined value at that point.

False.

For a function to be differentiable at a point, it must be continuous at that point.

False.

A function can be continuous at a point without being differentiable at that point.

'Differentiable at a point' implies 'continuous at a point', but 'continuous at a point' doesn't necessarily imply 'differentiable at a point'.

A function is smooth at a point if its graph does not have any corners or sudden changes of direction at that point.

The limit of as approaches , for and , is zero.