Here you will learn what is the rationalisation method to solve or find limits with examples.
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Rationalisation Method to Solve Limits
This method is particularly used when either the numerator or denominator or both involve expression consisting of square roots and substituting the value of x the rational expression takes the form \(0\over 0\), \(\infty\over \infty\).
Also Read : How to Solve Indeterminate Forms of Limits
Following examples illustrate the above method :
Example : Evaluate : \(\displaystyle{\lim_{x \to 0}}\) \(\sqrt{2 + x} โ \sqrt{2}\over x\).
Solution : When x = 0, the expression \(\sqrt{2 + x} โ \sqrt{2}\over x\) takes the form \(0\over 0\).
Rationalising the numerator we have,
\(\displaystyle{\lim_{x \to 0}}\) \((\sqrt{2 + x} โ \sqrt{2})(\sqrt{2 + x} + \sqrt{2})\over x(\sqrt{2 + x} + \sqrt{2})\)
= \(\displaystyle{\lim_{x \to 0}}\) \(2 + x โ 2\over x(\sqrt{2 + x} + \sqrt{2})\)
= \(\displaystyle{\lim_{x \to 0}}\) \(1\over x(\sqrt{2 + x} + \sqrt{2})\) = \(1\over 2\sqrt{2}\)
Example : Evaluate the limit : \(\displaystyle{\lim_{x \to 1}}\) [\({4 โ \sqrt{15x + 1}}\over {2 โ \sqrt{3x + 1}}\)]
Solution : \(\displaystyle{\lim_{x \to 1}}\) [\({4 โ \sqrt{15x + 1}}\over {2 โ \sqrt{3x + 1}}\)]
Rationalising the numerator and denominator both we have,
= \(\displaystyle{\lim_{x \to 1}}\) \({(4 โ \sqrt{15x + 1})(2 + \sqrt{3x + 1})(4 + \sqrt{15x + 1})}\over {(2 โ \sqrt{3x + 1})(4 + \sqrt{15x + 1})(2 + \sqrt{3x + 1})}\)
= \(\displaystyle{\lim_{x \to 1}}\) \((15 โ 15x)\over {3 โ 3x}\)\(\times\)\(2 + \sqrt{3x + 1}\over {4 + \sqrt{15x + 1}}\)
= \(5\over 2\)
