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प्रश्न
Evaluate the following :
`lim_(x -> ∞) [sqrt(x^2 + 4x + 16) - sqrt(x^2 + 16)]`
उत्तर
`lim_(x -> ∞)sqrt(x^2 + 4x + 16) - sqrt(x^2 + 16)`
= `lim_(x -> ∞) (sqrt(x^2 + 4x +16) - sqrt(x^2 + 16) sqrt(x^2 + 4x + 16) + sqrt(x^2 + 16))/(sqrt(x^2 + 4x + 16) + sqrt(x^2 + 16))`
= `lim_(x -> ∞) ((x^2 + 4x + 16 - x^2 - 16))/(sqrt(x^2 + 4x + 16) + sqrt(x^2 + 16))`
= `lim_(x -> ∞) (4x)/(sqrt(x^2 + 4x + 16) + sqrt(x^2 + 16))`
= `lim_(x -> ∞) ((4x)/x)/((sqrt(x^2 + 4x + 16) + sqrt(x^2 + 16))/x`
= `lim_(x -> ∞) ((4x)/x)/(sqrt((x^2 + 4x + 16)/x^2) + sqrt((x^2 + 16)/x^2))`
= `lim_(x -> ∞) 4/(sqrt(1 + 4/x + 16/x^2) + sqrt(1 + 16/x^2))`
= `4/(lim_(x -> ∞) sqrt(1 + 4/x + 16/x^2) + lim_(x -> ∞) sqrt(1 + 16/x^2)`
= `4/(sqrt(1 + 0 + 0) + sqrt(1 + 0)) ....[because lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`
= `4/2`
= 2
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