What is the best way to solve problem like these ↓ ? (Limits to infinity)

Question

Problems such as

When $x$ approaches positive infinity, find the limit of

$(3x-1)/(sqrt(x^2-6$

$(sqrt(4x^2+4x))/(4x+1)$

I know that you can solve by substituting values of $x$ as it gets closer to infinity, but is there a way to solve, like factorising, formula, or a general rule?

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Answer 1 · 2017-06-22T06:41:52

As a general rule convert a polynomial $f(x)$ to $g(1/x)$
(A) $3$ and (B) $1/2$

$Lt_(x->oo)(3x-1)/sqrt(x^2-6)$

$Lt_(x->oo)(3-1/x)/sqrt(1-6/x^2)$

= $3/sqrt1=3$
graph{(3x-1)/sqrt(x^2-6) [2.07, 19.55, -0.37, 8.37]}

$Lt_(x->oo)sqrt(4x^2+4x)/(4x+1)$

$Lt_(x->oo)sqrt(1+2/x)/(2+1/2x)$

= $1/2$
graph{sqrt(4x^2+4x)/(4x+1) [-0.152, 4.218, -0.51, 1.675]}

As a general rule convert a polynomial $f(x)$ to $g(1/x)$