Evaluation and Algebric of Limits - I

Various methods of evaluations of algebraic limits: To evaluate algebraic limits, we have the following methods:

i. Direct substitution method

ii. Factorization method

iii. Rationalisation method

iv. Using some standard results

v. Method of evaluating limits when variable tends to ∞ or - ∞

Direct substitution method: If by direct substitution of the point in the given expression we get a finite number, then the number obtained is the limit of the given expression.

Evaluate: 

Solution: 

Evaluate: 

Solution: 

Factorisation method: Consider . If by putting x = a, the rational function f(x)/g(x) takes the form0/0, ∞/∞ etc., then (x - a) is a factor of both f(x) and g(x). In such a case we factories the numerator and denominator and then cancel out the common factor (x – a). After cancelling out the common factor x - a we again put x = a in the given expression and see whether we get a meaningful number or not. This process is repeated till we get a meaningful number.

Rationalisation method: This method is generally used when one of numerator and denominator or both of them consist of expressions involving square roots.

Evaluate: 

Solution: We have,