Find the borders lim x-->∞ (sqrt (4x^2+9x^4)+5x^2)/((2x-1)^2+2x) lim x-->-1 (sqrt(3+2x)-1)/(sqrt(5+x)-2)lim x-->∞ (sqrt (4x^2+9x^4)+5x^2)/((2x-1)^2+2x) lim x-->-1...

Find the borders

lim x-->∞ (sqrt (4x^2+9x^4)+5x^2)/((2x-1)^2+2x)

lim x-->-1 (sqrt(3+2x)-1)/(sqrt(5+x)-2)

lim x-->∞ (sqrt (4x^2+9x^4)+5x^2)/((2x-1)^2+2x)

lim x-->-1 (sqrt(3+2x)-1)/(sqrt(5+x)-2)

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You need to force factor `x^4`  under the square root such that:

`lim_(x-gtoo) (sqrt(x^4(4/x^2 + 9)) + 5x^2)/((2x-1)^2+2x)`

You need to force factor `x^2`  to numerator and denominator such that:

`lim_(x-gtoo) (x^2(sqrt(4/x^2 + 9) + 5))/(x^2((2-1/x)^2+2/x))`

You need to reduce by `x^2`  such that:

`lim_(x-gtoo) (sqrt(4/x^2 + 9) + 5)/((2-1/x)^2+2/x)`

Substituting `oo`  for x in limit equation yields:

`(sqrt(4/oo + 9) + 5)/((2-1/oo)^2+2/oo) = (sqrt(9) + 5)/(2+0)`

`lim_(x-gtoo) (sqrt(4/x^2 + 9) + 5)/((2-1/x)^2+2/x)= (3+5)/2`

`lim_(x-gtoo) (sqrt(4/x^2 + 9) + 5)/((2-1/x)^2+2/x) = 8/2`

`lim_(x-gtoo) (sqrt(4/x^2 + 9) + 5)/((2-1/x)^2+2/x) = 4`

Hence, evaluating the limit yields`lim_(x-gtoo) (x^2(sqrt(4/x^2 + 9) + 5))/(x^2((2-1/x)^2+2/x)) = 4.`

You need to evaluate the second limit, hence, you need to substitute -1 for x in equation under limit such that:

`lim_(x-gt-1)(sqrt(3+2x)-1)/(sqrt(5+x)-2) = (sqrt(3-2)-1)/(sqrt(5-1)-2)`

`lim_(x-gt-1)(sqrt(3+2x)-1)/(sqrt(5+x)-2) = (1-1)/(2-2) = 0/0`

The limit is indeterminate and since the type of indetermination is `0/0` , you may use l'Hospital's theorem such that:

`lim_(x-gt-1)(sqrt(3+2x)-1)/(sqrt(5+x)-2) = lim_(x-gt-1)((sqrt(3+2x)-1)')/((sqrt(5+x)-2)') `

`lim_(x-gt-1)((sqrt(3+2x)-1)')/((sqrt(5+x)-2)') = lim_(x-gt-1)(2/(2sqrt(3+2x)))/(1/(2sqrt(5+x)))`

You need to substitute -1 for x such that:

`lim_(x-gt-1)(1/(sqrt(3+2x)))/(1/(2sqrt(5+x))) = (1/(sqrt(3-2)))/(1/(2sqrt(5-1)))`

`lim_(x-gt-1)(1/(sqrt(3+2x)))/(1/(2sqrt(5+x))) = 1/(1/4)`

`lim_(x-gt-1)(1/(sqrt(3+2x)))/(1/(2sqrt(5+x))) = 4`

Hence, evaluating the limit of the function `lim_(x-gt-1)(sqrt(3+2x)-1)/(sqrt(5+x)-2)`  yields `lim_(x-gt-1)(sqrt(3+2x)-1)/(sqrt(5+x)-2) = 4.`

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