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lim x → 1x-12...

Question

$\lim_{x \to 1} \frac{(\sqrt{x} - 1) (2 x - 3)}{2 x^{2} + x - 3} is$

$(a) \frac{1}{10} (b) - \frac{1}{10}$

(c) 1

(d) none of these

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Solution

$i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{(\sqrt{x} - 1) (2 x - 3)}{2 x^{2} + x - 3} i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{(\sqrt{x} - 1) (2 x - 3)}{2 x^{2} + 3 x - 2 x - 3} i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{(\sqrt{x} - 1) (2 x - 3)}{(2 x^{2} - 2 x) + (3 x - 3)} i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{(\sqrt{x} - 1) (2 x - 3)}{2 x (x - 1) + 3 (x - 1)} i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{(\sqrt{x} - 1) (2 x - 3)}{(x - 1) (2 x + 3)} i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{(\sqrt{x} - 1) (2 x - 3)}{(\sqrt{x} - 1) (\sqrt{x} + 1) (2 x + 3)} i . e . \begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{2 x - 3}{(\sqrt{x} + 1) (2 x + 3)} i . e . (\begin{matrix} \lim \\ x \to 1 \end{matrix} 2 x - 3) (\begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{1}{\sqrt{x} + 1}) (\begin{matrix} \lim \\ x \to 1 \end{matrix} \frac{1}{2 x + 3}) i . e . (- 1) \times \frac{1}{2} \times \frac{1}{5} = \frac{- 1}{10}$

Hence, the correct answer is option B.

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