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# limit-continuity-3

2) __Algebraic Limits: __

A) Factorization process of finding limit.

If direct substitution of x=a in a rational function takes useless form like _{ } then a factor of (x-a) can be cancelled from both numerator and denominator. Now again use direct substitution to see if limit can be evaluated.

Illustration 7:

Find_{ } ?

Solution:

_{ }

B) Using the result _{ }

Illustration 8: Evaluate_{ } ?

Solution:

_{ }

C) Using rationalization:

Some factor which creates _{ } form is cancelled by rationalization and limit is evaluated.

Illustration 9: Evaluate_{ } ?

Solution:

_{ }

_{ }

D) Limits where x®

We write down expression as a rational function and then divide each term by highest power of x obtained from numerator and denominator.

Illustration 10: Find the value of_{ } ?

Solution:

_{ }

3) __Trigonometric Limits:__

The trigonometric limits are those which involve trigonometric ratios. The use of the result _{ } is a integral part of evaluation of such limits.

Illustration 11: Evaluate_{ } ?

_{ }

= 1´ p ´ 1

= p.

4) __Exponential and Logarithmic Limits:__

These are the limits which involve use of logarithmic and exponential functions. Some of the standard limits help us in solving these limits.

Illustration 12:

Find the following limits _{ }

Solution:

_{ }