Question

What is the limit of `(3x^2) / (x^2+2x)` as x approaches infinity?

Answer 1

`3`

Explanation:

divide terms on numerator/denominator by the highest power of x, that is `x^2`

`((3x^2)/x^2)/(x^2/x^2+(2x)/x^2)=3/(1+2/x)`

`rArrlim_(xtooo)(3x^2)/(x^2+2x)`

`=lim_(xtooo)3/(1+2/x)`

`=3/(1+0)`

`=3`

Answer 2

`lim_(x->oo) (3x^2)/(x^2+2x) =3`

Explanation:

We can manipulate the limit as follows

`lim_(x->oo) (3x^2)/(x^2+2x) = lim_(x->oo) (3x^2)/(x^2+2x) *(1/x^2)/(1/x^2)`
`" "= lim_(x->oo) (3)/(1+2/x)`
`" "= 3`

As `2/x rarr 0` as `x rarr oo`