is continuous at every point in the domain of f . Thus f is a continuous function. We take this opportunity to explain the concept of infinity . This we do by analysing the function f ( x ) = x near x = .
To carry out this analysis we follow the usual trick of finding the value of the function at real numbers close to . Essentially we are trying to find the right hand limit of f at . We tabulate this in the following (Table . ).
= = n We observe that as x gets closer to from the right, the value of f ( x ) shoots up higher. This may be rephrased as: the value of f ( x ) may be made larger than any given number by choosing a positive real number very close to . In symbols, we write = + ∞ (to be read as: the right hand limit of f ( x ) at is plus infinity). We wish to emphasise that + ∞ is NOT a real number and hence the right hand limit of f at does not exist (as a real number).
Similarly, the left hand limit of f at may be found. The following table is self explanatory. Table . – – .
– . – – – – – – – – n f ( x ) – – . ... – – – – – n From the Table .
, we deduce that the value of f ( x ) may be made smaller than any given number by choosing a negative real number very close to . In symbols, we write = −∞ (to be read as: the left hand limit of f ( x ) at is minus infinity).