lim _{x rightarrow 1} frac{(2 x-3)(sqrt{x}-1) | vedclass

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(B) Given limit: $\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}$ Factorize the denominator: $2x^2+x-3 = (2x+3)(x-1)$ Substitute this i

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$-\frac{1}{10}$

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(B) Given limit: $\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3}$ Factorize the denominator: $2x^2+x-3 = (2x+3)(x-1)$ Substitute this into the limit: $\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{(2 x+3)(x-1)}$ Multiply the numerator and denominator by $(\sqrt{x}+1)$: $\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)(\sqrt{x}+1)}{(2 x+3)(x-1)(\sqrt{x}+1)}$ Simplify using $(\sqrt{x}-1)(\sqrt{x}+1) = (x-1)$: $\lim _{x \rightarrow 1} \frac{(2 x-3)(x-1)}{(2 x+3)(x-1)(\sqrt{x}+1)}$ Cancel $(x-1)$ from numerator and denominator: $\lim _{x \rightarrow 1} \frac{2 x-3}{(2 x+3)(\sqrt{x}+1)}$ Evaluate the limit by substituting $x=1$: $\frac{2(1)-3}{(2(1)+3)(\sqrt{1}+1)} = \frac{-1}{5(2)} = -\frac{1}{10}$

$\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3} = $

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