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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता ११ वी

Evaluate the following : limx→∞[x4+4x2-x2] - Mathematics and Statistics

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प्रश्न

Evaluate the following :

`lim_(x -> ∞) [sqrt(x^4 + 4x^2) - x^2]`

बेरीज

उत्तर

Let L = `lim_(x -> ∞) [sqrt(x^4 + 4x^2) - x^2]`

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

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

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

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

Dividing numerator and denominator by x2, we get,

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

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

= `4/(sqrt(1 + 4 xx 0) + 1)   ...[because lim_(x -> ∞) 1/x^"n" = 0  "if n" > 0]`

= 2

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Limit at Infinity
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 7: Limits - Exercise 7.7 [पृष्ठ १५७]

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