Limit to Infinity with Sqrt in Denom.(Is this correct?)

[Ordain]

Lim
x→∞ $\frac{7x^2-14x+7}{\sqrt{2x^4-4x^3+x+7}}$

Normally wouldn't have an issue here, just slightly confused by the sqrt.

Attempted solution:

$\frac{7x^2-14x+7}{\sqrt{2x^4-4x^3+x+7}}*\frac{x^-2}{x^-2}$

Yields $\frac{7}{\sqrt{2}}$

Is this correct?

Similarly:

lim
x→-∞
$\frac{x^3+x+1}{\sqrt{2x^6+x^3+x+3}}$

Yields $\frac{1}{\sqrt{2}}$

and
lim
x→-∞
$\frac{x^2+x+1}{\sqrt{2x^4+x^3+x+3}}$

Yields $\frac{1}{\sqrt{2}}$

 

[dextercioby]

This is correct. The proof to all of that is to forcibly divide both the numerator and the denominator by x to the power of the highest term in the non-square rooted expression and then take the limit to infinity.

 

[HallsofIvy]

By the way, use x^{-2} to get $x^{-2}$. x^-2 puts only the "-" in the exponent: $x^-2$.