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

$10^{4x}\to10^{\text{large negative number}}=\dfrac1{10^{\text{large positive number}}}$

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

$\displaystyle\lim_{x\to-\infty}2\cdot10^{4x}=0$