Homework Help

Evaluate the limit((Sqrt(4+x))-2)/(x)

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hellome234 | Salutatorian

Posted May 10, 2012 at 5:46 AM via web

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Evaluate the limit

((Sqrt(4+x))-2)/(x)

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thilina-g | College Teacher | (Level 1) Educator

Posted May 10, 2012 at 6:15 AM (Answer #1)

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You have not included the limit. But by looking at the question I can see that the intended limit is 0.

`lim_(x-gt0)(sqrt(4+x)-2)/x`

If you try to evaluate the limit straight away, you will see that it will result in 0/0 situation which is indertminate. To remove this, we can multiply both numerator and denominator by `(sqrt(4+x)+2)` .

`lim_(x-gt0)((sqrt(4+x)-2)(sqrt(4+x)+2))/(x(sqrt(4+x)+2))`

This would give,

`lim_(x-gt0)((4+x)-2^2)/(x(sqrt(4+x)+2))`

`lim_(x-gt0)x/(x(sqrt(4+x)+2))`

This simplifies into,

`lim_(x-gt0)1/(sqrt(4+x)+2)`

Now you can evaluate the limit easily,

`lim_(x-gt0)1/(sqrt(4+x)+2) = 1/(2+2) = 1/4`

 

Therefore,

`lim_(x-gt0)(sqrt(4+x)-2)/x = 1/4`

 

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