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Evaluate: limx→2x2-43x-2-x+2 - Mathematics

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Sum

Evaluate: `lim_(x -> 2) (x^2 - 4)/(sqrt(3x - 2) - sqrt(x + 2))`

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Solution

Given that `lim_(x -> 2) (x^2 - 4)/(sqrt(3x - 2) - sqrt(x + 2))`

Rationalizing the denominator, we get

= `lim_(x -> 2) ((x - 2)(x + 2) [sqrt(3x - 2) + sqrt(x + 2)])/([sqrt(3x - 2) - sqrt(x + 2)][sqrt(3x - 2) + sqrt(x + 2)])`

= `lim_(x -> 2) ((x - 2)(x + 2)[sqrt(3x - 2) + sqrt(x + 2)])/(3x - 2 - x - 2)`

= `lim_(x -> 2) ((x - 2)(x + 2)[sqrt((3x - 2)) + sqrt(x + 2)])/(2x - 4)`

= `lim_(x -> 2) ((x - 2)(x + 2) [sqrt((3x - 2)) + sqrt(x + 2)])/(2(x - 2))`

= `lim_(x -> 2) ((x + 2)[sqrt(3x - 2) + sqrt(x + 2)])/2` 

Taking limits, we have

`= ((2 + 2)[sqrt(6 - 2) + sqrt(2 + 2)])/2`

= `(4[2 + 2])/2`

= `(4 xx 4)/2`

= 8

Concept: Concept of Limits - Limits of Polynomials and Rational Functions
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APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 13 Limits and Derivatives
Exercise | Q 8 | Page 240

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