Maharashtra State BoardHSC Science (General) 11th
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Evaluate the following : limx→∞[x2+5-x2-3x2+3-x2+1] - Mathematics and Statistics

Sum

Evaluate the following :

`lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1))]`

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Solution

`lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1))]`

= `lim_(x -> ∞) [(sqrt(x^2 + 5) - sqrt(x^2 - 3))/(sqrt(x^2 + 3) - sqrt(x^2 + 1)) xx (sqrt(x^2 + 5) + sqrt(x^2 - 3))/(sqrt(x^2 + 3) + sqrt(x^2 + 1)) xx (sqrt(x^2 + 3) + sqrt(x^2 + 1))/(sqrt(x^2 + 5) + sqrt(x^2 - 3))]`

= `lim_(x -> ∞)[((x^2 + 5) - (x^2 - 3))/((x^2 + 3) - (x^2 + 1)) xx(sqrt(x^2 + 3) + sqrt(x^2 + 1))/(sqrt(x^2 + 5) + sqrt(x^2 - 3))]`

= `lim_(x -> ∞) (8(sqrt(x^2 + 3) + sqrt(x^2 + 1)))/(2(sqrt(x^2 + 5) + sqrt(x^2 - 3))`

= `4lim_(x -> ∞) [((sqrt(x^2 + 3) + sqrt(x^2 + 1))/x)/((sqrt(x^2 + 5) + sqrt(x^2 - 3))/x)]   ...[("Divide numerator and"),("denominator by"  x)]`

= `4lim_(x -> ∞) [((sqrt(x^2 + 3))/x + (sqrt(x^2 + 1))/x)/(sqrt(x^2 + 5)/x + sqrt(x^2 - 3)/x)]`

= `(4lim_(x -> ∞)[sqrt((x^2 + 3)/x^2) + sqrt((x^2 + 1)/x^2)])/(lim_(x -> ∞)[sqrt((x^2 + 5)/x^2) + sqrt((x^2 - 3)/x^2)])`

= `(4(lim_(x -> ∞) sqrt(1 + 3/x^2) + lim_(x -> ∞) sqrt(1 + 1/x^2]))/(lim_(x -> ∞) sqrt(1 + 5/x^2) + lim_(x -> ∞) sqrt(1 - 3/x^2)`

= `(4(sqrt(1 + 0) + sqrt(1 + 0)))/(sqrt(1 + 0) + sqrt(1 - 0))  ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`

= `(4(1 + 1))/(1 + 1)`

= 4

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