Question

Question #72096

Answer 1

The limit equals `3`.

Explanation:

Divide by the highest power, which is `x^2`.

`L = lim_(x->oo) ((3x^2 + 3)/x^2)/((x^2 - 7x +9 )/x^2)`

`L = lim_(x->oo) (3 + 3/x^2)/(1 - 7/x + 9/x^2)`

We know that `lim_(x->oo) 1/x = 0`, therefore:

`L = 3/1`

`L = 3`

Hopefully this helps!

Answer 2

Because the expression evaluated at `oo` yields the indeterminate form `oo/oo`, one should use L'Hôpital's rule.

Explanation:

Given: `lim_(x to oo)(3x^2+3)/(x^2-7x+9)`

To apply L'Hôpital's rule, differentiate the numerator and the denominator:

`lim_(xtooo)((d(3x^2+3))/dx)/((d(x^2-7x+9))/dx)`

`lim_(xtooo)(6x)/(2x-7)`

Because the above, also, yields the indeterminate form `oo/oo`, we can apply the rule, again:

`lim_(xtooo)((d(6x))/dx)/((d(2x-7))/dx)`

`lim_(xtooo)6/2 = 3`

L'Hôpital's rule states that the original limit goes to the same value:

`lim_(x to oo)(3x^2+3)/(x^2-7x+9) = 3`