Question

Can someone explain to me how do I do this?

This is for AP Calculus ​

Answer 1

1. As
`x\to+\infty`, the expression
`x^3` will remain positive and veer off to positive infinity as well. The constant term and coefficient on
`x^3` don't affect this result:

`\displaystyle\lim_(x\to\infty)(0.25x^3+3)=\infty`

In the opposite direction,
`x\to-\infty` means
`x` will be negative, so
`x^3` will also be negative. Then

`\displaystyle\lim_(x\to-\infty)(0.25x^3+3)=-\infty`

2. As
`x\to+\infty`, the exponent
`4x` will also go to positive infinity, so that

`\displaystyle\lim_(x\to\infty)2\cdot10^(4x)=\infty`

In the opposite direction,
`4x` would veer off to negative infinity. Then

`10^(4x)\to10^{\text{large negative number}}=\frac1{10^{\text{large positive number}}}`

and as
`x` gets larger in magnitude, this would force the expression to converge to 0, so that

`\displaystyle\lim_(x\to-\infty)2\cdot10^(4x)=0`