Question

Hello’ I need help with this practice problem in calculus

Answer 1

Step-by-step explanation:

Given the below;

`\lim _(x\to-\infty)(2x^3-2x)=-\infty`

We're to determine if the above is true or not.

Let us pick two negative values for x and see what we'll have.

When x = -1, we'll have;

`2(-1)^3-2(-1)=2(-1)+2=-2+2=0`

When x = -5, we'll have;

`2(-5)^3-2(-5)=2(-125)+10=-250+10=-240`

We can see from the above, that x tends to negative infinity, the function keeps getting smaller, also tending to negative infinity, therefore we can say that the given limit statement is true.

Given the below;

`\lim _(x\to\infty)(-2x^4+6x^3-2x)=-\infty`

To determine if the above is true or not, let's pick any two values of x and see what we'll have.

We'll only consider the term with the highest degree as this term can also show us what will happen to the polynomial just as the whole can;

When x = 1;

`-2(1)^4=-2(1)=-2`

When x = 10;

`-2(10)^4=-2(10000)=-20000`

We can see from the above, that x tends to positive infinity, the function keeps getting smaller, tending to negative infinity, therefore we can say that the given limit statement is true.

Given the below;

`\lim _(x\to\infty)(9x^5-6x^3-x)=-\infty`

To determine if the above is true or not, let's pick any two values of x and see what we'll have.

We'll only consider the term with the highest degree as this term can show us what will happen to the polynomial just as the whole function;

When x = 1;

`9(1)^5=9(1)=9`

When x = 10;

`9(10)^5=9(100000)=900000`

We can see from the above, that x tends to positive infinity, the function keeps getting larger, tending to positive infinity, therefore we can say that the given limit statement is false.