Evaluate the following limits: limx→2[x3-8x+2-3x-2] - Mathematics and Statistics

Evaluate the following limits: `lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))]`

Sum

`lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2))]`

= `lim_(x -> 2) [(x^3 - 8)/(sqrt(x + 2) - sqrt(3x - 2)) xx (sqrt(x + 2) + sqrt(3x - 2))/(sqrt(x + 2) + sqrt(3x - 2))]`

= `lim_(x -> 2) ((x^3 - 8)(sqrt(x + 2) + sqrt(3x - 2)))/((x + 2) - (3x - 2)`

= `lim_(x -> 2) ((x^3 - 8)(sqrt(x + 2) + sqrt(3x - 2)))/(-2x + 4)`

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

= `lim_(x -> 2) ((x^2 + 2x + 4)(sqrt(x + 2) + sqrt(3x - 2)))/(-2) ...[("As" x -> 2"," x ≠ 2),(therefore x - 2 ≠)]`

= `-1/2 lim_(x -> 2) (x^2 + 2x + 4) xx lim_(x -> 2) (sqrt(x + 2) + sqrt(3x - 2))`

= `-1/2 xx [(2)^2 + 2(2) + 4] xx (sqrt(2 + 2) + sqrt(3(2) - 2))`

= `-1/2 xx 12 xx (2 + 2)`

= – 6 x 4
= – 24

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Chapter 7: Limits - EXERCISE 7.3 [Page 103]

Chapter 7 Limits
EXERCISE 7.3 | Q III. 4. | Page 103

Evaluate the following limits: `lim_(x -> 2)[(sqrt(2 + x) - sqrt(6 - x))/(sqrt(x) - sqrt(2))]`

Evaluate the following limits: `lim_(x -> "a") [(sqrt("a" + 2x) - sqrt(3x))/(sqrt(3"a" + x) - 2sqrt(x))]`

Evaluate the following limits: `lim_(x -> 2)[(x^2 - 4)/(sqrt(x + 2) - sqrt(3x - 2))]`

``Evaluate the following limits: `lim_(x -> 4) [(x^2 + x - 20)/(sqrt(3x + 4) - 4)]`

Evaluate the following limits: `lim_(y -> 2) [(2 - y)/(sqrt(3 - y) - 1)]`

Evaluate the following limit:

`lim_(x -> 0)[(sqrt(6 + x + x^2) - sqrt(6))/x]`