`lim_(x->oo) (sqrt(x^2+4x) - sqrt(x^2 - 6x))`What is the limit plus infinity of the square root of x squared plus 4x minus the square root of x squared minus 6x?I dnt stop falling on indefinite...

`lim_(x->oo) (sqrt(x^2+4x) - sqrt(x^2 - 6x))`

What is the limit plus infinity of the square root of x squared plus 4x minus the square root of x squared minus 6x?I dnt stop falling on indefinite form

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the following limit, hence, you need to substitute `oo` for `x` such that:

`lim_(x->oo) (sqrt(x^2+4x) - sqrt(x^2 - 6x)) = sqrt(oo^2+oo) - sqrt(oo^2-oo) = oo - oo`

Since the result is indeterminate and since the expression under limit involves radicals, you need to multiply and subtract by its conjugate `(sqrt(x^2+4x) + sqrt(x^2 - 6x))` such that:

`lim_(x->oo) ((sqrt(x^2+4x) - sqrt(x^2 - 6x))(sqrt(x^2+4x) + sqrt(x^2 - 6x)))/(sqrt(x^2+4x) + sqrt(x^2 - 6x))`

You should convert the product to numerator into a difference of squares, such that:

`lim_(x->oo) (x^2 + 4x - (x^2 - 6x))/(sqrt(x^2+4x) + sqrt(x^2 - 6x))`

`lim_(x->oo) (x^2 + 4x - x^2 + 6x)/(sqrt(x^2+4x) + sqrt(x^2 - 6x))`

Reducing like terms yields:

`lim_(x->oo) (10x)/(sqrt(x^2+4x) + sqrt(x^2 - 6x))`

You need to force the factor `x^2` under the radicals, such that:

`lim_(x->oo) (10x)/(sqrt(x^2(1+4/x)) + sqrt(x^2(1 - 6/x)))`

You need to remember that `sqrt(x^2) = +-x` , but, since `x->oo` yields `sqrt(x^2) = x` , such that:

`lim_(x->oo) (10x)/(xsqrt(1+4/x) + xsqrt(1 - 6/x))`

You need to factor out x such that:

`lim_(x->oo) (10x)/(x(sqrt(1+4/x) + sqrt(1 - 6/x)))`

Reducing duplicate factors yields:

`lim_(x->oo) 10/((sqrt(1+4/x) + sqrt(1 - 6/x))) `

Since `lim_(x->oo) 1/x = 0` yields:

`lim_(x->oo) 10/((sqrt(1+4/x) + sqrt(1 - 6/x))) = 10/(sqrt(1 + 0) + sqrt(1+0))`

`lim_(x->oo) 10/((sqrt(1+4/x) + sqrt(1 - 6/x))) = 10/2`

`lim_(x->oo) 10/((sqrt(1+4/x) + sqrt(1 - 6/x))) = 5`

Hence, evaluating the given limit yields `lim_(x->oo) (sqrt(x^2+4x) - sqrt(x^2 - 6x)) = 5.`

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