Question

Use the limit theorem and the properties of limits to find the limit.

Answer 1

`b. (2)/(3)`

Explanation:

The given limit is

`\lim_(x \to \infty) (-2x^3+7x+2)/(-3x^3-x+2)`

We divide the numerator and the denominator by
`x^3`

`\lim_(x \to \infty) ((-2x^3)/(x^3)+(7x)/(x^3)+(2)/(x^3))/((-3x^3)/(x^3)-(x)/(x^3)+(2)/(x^3))`

This simplifies to;

`\lim_(x \to \infty) (-2+(7)/(x^2)+(2)/(x^3))/(-3-(1)/(x^3)+(2)/(x^2))`

Apply the following limit property;

As
`x\rightarrow -\infty, (c)/(x^n) \rightarrow 0`, where c is a constant.

This implies that;

`\lim_(x \to \infty) (-2+(7)/(x^2)+(2)/(x^3))/(-3-(1)/(x^3)+(2)/(x^2))=(-2+0+0)/(-3-0+0)`

`\therefore \lim_(x \to \infty) (-2x^3+7x+2)/(-3x^3-x+2)=(2)/(3)`