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: \(\underset{x\to 1}{\mathop{\lim }}\,\) (3x² + 4x + 5)
Solution: \(\underset{x\to 1}{\mathop{\lim }}\,\) (3x² + 4x + 5) = 3 (1)² + 4 (1) + 5 = 12
Evaluate: \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos x}{1+\sin x}\)
Solution: \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos x}{1+\sin x}=\frac{\cos 0}{1+\sin 0}=1\)
Factorisation method: Consider \(\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}\). 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: \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{m}}-{{a}^{m}}}{{{x}^{n}}-{{a}^{n}}}\).
Solution: We have,
\(\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{m}}-{{a}^{m}}}{{{x}^{n}}-{{a}^{n}}}\),
= \(\underset{x\to a}{\mathop{\lim }}\,\left[ \frac{{{x}^{m}}-{{a}^{m}}}{x-a}.\frac{x-a}{{{x}^{n}}-{{a}^{n}}} \right]\),
= \(\underset{x\to a}{\mathop{\lim }}\,\left[ \frac{{{x}^{m}}-{{a}^{m}}}{x-a}\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a} \right]\),
= \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{m}}-{{a}^{m}}}{x-a}\div \underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a}\),
\(m{{a}^{m-1}}\div n{{a}^{n-1}}=\frac{m}{n}{{a}^{m-n}}\).