Question

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

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?

Answer 1

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

Explanation:

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

• let us divide numerator and denominator by `x` and we get

`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)`

• dividing numerator and denominator by `2x` and we get

`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)`