Question-134006

Question Number 134006 by mohammad17 last updated on 26/Feb/21

Answered by malwan last updated on 26/Feb/21

(1) \underset{x \to 4}{l i m} \frac{2 x - 5}{3} = \frac{2 \times 4 - 5}{3} = 1

(2) \underset{x \to - 3^{+}}{l i m} \frac{\sqrt{x + 4} - \sqrt{- x - 2}}{\sqrt{x + 3}} \times \frac{\sqrt{x + 4} + \sqrt{- x - 2}}{\sqrt{x + 4} + \sqrt{- x - 2}}

= \underset{x \to - 3^{+}}{l i m} \frac{2 \sqrt{x + 3} \sqrt{x + 3}}{\sqrt{x + 3} (\sqrt{x + 4} + \sqrt{- x - 2})} = 0

b u t \underset{x \to - 3^{-}}{l i m} {d o e s n}^{'} t e x i s t

(3) \underset{x \to \sqrt{2}}{l i m} \frac{2 - x^{2}}{\sqrt{2} - x} = \underset{x \to \sqrt{2}}{l i m} \frac{(\sqrt{2} + x) (\sqrt{2} - x)}{\sqrt{2} - x} = 2 \sqrt{2}

(4) \underset{x \to 0}{l i m} \frac{s i n x - c o s x s i n x}{x^{2}}

= \underset{x \to 0}{l i m} \frac{s i n x (1 - c o s x)}{x^{2}} = \underset{x \to 0}{l i m} \frac{s i n x \times 2 {s i n}^{2} \frac{x}{2}}{x^{2}}

= 0

(5) \underset{x \to 1}{l i m} \frac{\frac{1}{\sqrt{x}} - 1}{1 - x} = \underset{x \to 1}{l i m} \frac{\frac{1 - \sqrt{x}}{\sqrt{x}}}{(1 + \sqrt{x}) (1 - \sqrt{x})}

= \frac{1}{2}

(6) \underset{x \to \infty}{l i m} \frac{- x^{7} + 3 x^{5} + 4}{7 x^{7} - 6 x^{6} + 5 x^{5} + 3} = - \frac{1}{7}

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