Now we discuss the various methods used in obtaining limits. Each method will be accompanied by some examples illustrating that method.

**(A) DIRECT SUBSTITUTION**

This already finds mention at the start of the current section, where we saw that for a continuous function, the limit can be obtained by direct substitution.

This is because, by definition of a continuous function (at ):

Hence, for example, all polynomial limits can be evaluated by direct substitution.

Some examples make all this clear:

(i) | |

(ii) | |

(iii) | |

(iv) | |

(v) | |

(vi) |

**(B) FACTORIZATION**

We saw an example of this method in evaluating .

(i) |

In such forms, the limit is indeterminate due to a certain factor occuring in the expression (For example, in the limit above, occurs in both the numerator and denominator and makes the limit indeterminate, of the form ) . Factorization leads to cancellation of that common factor and reduction of the limit to a determinate form.

Note that this limit is also of the form (whose limit is )

(ii) |

(iii) |

We see that upon substitution of , both the numerator and denominator become . Hence, is a factor of both the numerator and denominator (Factor theorem)

(iv) |

Factorization leads to

Cancelling out from the numerator and the denominator. | |