Find the limit using limit principles for infinite limits as x goes to infinity for (2x^2 +1)/(9x^4 + 2)^-1/2

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I think you have made mistake typing, this should be (2x^2+1)/(9x^4+2)^(1/2), not minus -1/2. Assuming that I will do the sum.

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

Now if you try to evaluate the limit straight away, you would get the answer as `oo/oo` , which is indeterminate. We can remove this by dividing both numerator and denominator by `x^2.`

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

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

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

Now we can evaluate the limit.

`lim_(x-gtoo)(2+1/x^2)/sqrt(9+2/x^4) = (2+0)/sqrt(9+0)`

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

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

Therefore,

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

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