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Question
`lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(2x^2 + x - 3)` is ______.
Options
`1/10`
`(-1)/10`
1
None of these
Solution
`lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(2x^2 + x - 3)` is `(-1)/10`.
Explanation:
Given `lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(2x^2 + 3x - 2x - 3)`
= `lim_(x -> 1) ((sqrt(x) - 1)(2x - 3))/(x(2x + 3) - 1(2x + 3))`
= `lim_(x -> 1)((sqrt(x) - 1)(2x - 3))/((x - 1)(2x + 3))`
= `lim_(x -> 1) ((sqrt(x) - 1)(sqrt(x) + 1)(2x - 3))/((x - 1)(sqrt(x) + 1)(2x + 3))`
= `lim_(x + 1) ((x - 1)(2x - 3))/((x - 1)(sqrt(x) + 1)(2x + 3))`
= `lim_(x -> 1) (2x - 3)/((sqrt(x) + 1)(2x + 3))`
Taking limit we have
= `(2(1) - 3)/((sqrt(1) + 1)(2 xx 1 + 3))`
= `(-1)/(2 xx 5)`
= `(-1)/10`
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