Question

PLease help me solve this

Answer 1

`y = \infty`

Explanation:

We are given

`f(x) = \left((x^7+3x^4-15)/(x^5+3x)\right)`

and asked to find

`\lim _(x\to \pm \infty ) f(x)`

Note in the numerator x⁷ is the dominating term. So the behavior as

`x \rightarrow \pm \infty` is controlled by this term

Similarly in the denominator, x⁵ is the dominating term and the behaviour as
`x \rightarrow \pm \infty` is controlled by this term

So

`\lim _(x\to \pm \infty ) f(x) \\\\= \lim _(x\to \pm \infty ) \left((x^7+3x^4-15)/(x^5+3x)\right)\\\\\\= \lim _(x\to \pm \infty ) (x^7)/(x^5)`

`\text{$ (x^7)/(x^5) $ simplifies to $ x^2 $}`

`\lim _(x\to \pm \infty ) \left((x^7+3x^4-15)/(x^5+3x)\right)\\\\\\= \lim _(x\to \pm \infty ) (x^7)/(x^5)\\\\\\= \lim _(x\to \pm \infty ) x^2\\\\= \infty`

Answer

`y = \infty`