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प्रश्न
Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1
= `-1/3` for x = 1, at x = 1
उत्तर
Given `f(1) = (-1)/3`
Consider
`lim_(x ->1) f(x) = lim_(x ->1) (3 - sqrt(2x + 7))/(x - 1)`
= `lim_(x ->1) (3 - sqrt(2x + 7))/(x - 1) xx (3 + sqrt(2x + 7))/(3 + sqrt(2x + 7))`
= `lim_(x ->1) (9 - 2x - 7)/((x - 1) (3 + sqrt(2x + 7))`
= `lim_(x ->1) (2 - 2x)/((x - 1) (3 + sqrt(2x + 7))`
= `lim_(x ->1) (-2(x - 1))/((x - 1) (3 + sqrt(2x + 7))`
= `lim_(x ->1) (-2)/(3 + sqrt(2x + 7)`
(`therefore x -> 1, ( x- 1) ≠ 0)`
= `(-2)/(3 + sqrt(2x + 7)`
= `(-2)/(3 + sqrt9)`
= `(-2)/(3 + 3)`
= `(-2)/6`
= `(-1)/3`
`lim_(x ->1) f(x) = f(1)`
Function is continuous at x = 1
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