Find limits: 1.) lim x-->∞ (5+sqrt(x^2+5))/(x-6) 2.) lim x-->∞ (10x^3-4)/(x^2+2x-6) 3.) lim x-->∞ (sqrt^3(x^3-4x^2-7x-5))/(sqrt(x^2-7x)+9)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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

`lim_(x-gtoo) (5+sqrt(x^2(1+5/x^2)))/(x(1-6/x))`

You need to remember that `sqrt(x^2) = +-x`  but you need to keep the positive value since x approaches to `+oo` .

`lim_(x-gtoo) (5+xsqrt(1+5/x^2))/(x(1-6/x))`

You need to force factor x to numerator  such that:

`lim_(x-gtoo) x(5/x+sqrt(1+5/x^2))/(x(1-6/x))`

Reducing by x yields:

`lim_(x-gtoo) (5/x+sqrt(1+5/x^2))/(1-6/x)`

You need to substitute `oo ` for x such that:

`lim_(x-gtoo) (5/x+sqrt(1+5/x^2))/(1-6/x) = (5/oo+sqrt(1+5/oo))/(1-6/oo)`

`lim_(x-gtoo) (5/x+sqrt(1+5/x^2))/(1-6/x) = 1`

Hence, evaluating the limit `lim_(x-gtoo) (5+sqrt(x^2+5))/(x-6)`  yields `lim_(x-gtoo) (5+sqrt(x^2+5))/(x-6) = 1` .

2) You need to force factor `x^3`  to numerator and `x^2`  to denominator such that:

`lim_(x-gtoo) (10x^3-4)/(x^2+2x-6) = lim_(x-gtoo) (x^3(10-4/x^3))/(x^2(1+2/x-6/x^2))`

`lim_(x-gtoo) (10x^3-4)/(x^2+2x-6) = lim_(x-gtoo) (x(10-4/x^3))/((1+2/x-6/x^2))`

You need to substitute `oo`  for x such that:

`lim_(x-gtoo) (x(10-4/x^3))/((1+2/x-6/x^2)) = (oo(10-4/oo))/((1+2/oo-6/oo))`

`lim_(x-gtoo) (x(10-4/x^3))/((1+2/x-6/x^2)) = (oo(10-0))/((1+0-0))`

`lim_(x-gtoo) (x(10-4/x^3))/((1+2/x-6/x^2)) = (oo*10)/1`

`lim_(x-gtoo) (x(10-4/x^3))/((1+2/x-6/x^2)) = oo`

Hence, evaluating the limit `lim_(x-gtoo) (10x^3-4)/(x^2+2x-6)`  yields `lim_(x-gtoo)(10x^3-4)/(x^2+2x-6) = oo.`

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