1. As [tex]x\to+\infty[/tex], the expression [tex]x^3[/tex] will remain positive and veer off to positive infinity as well. The constant term and coefficient on [tex]x^3[/tex] don't affect this result:
[tex]\displaystyle\lim_{x\to\infty}(0.25x^3+3)=\infty[/tex]
In the opposite direction, [tex]x\to-\infty[/tex] means [tex]x[/tex] will be negative, so [tex]x^3[/tex] will also be negative. Then
[tex]\displaystyle\lim_{x\to-\infty}(0.25x^3+3)=-\infty[/tex]
2. As [tex]x\to+\infty[/tex], the exponent [tex]4x[/tex] will also go to positive infinity, so that
[tex]\displaystyle\lim_{x\to\infty}2\cdot10^{4x}=\infty[/tex]
In the opposite direction, [tex]4x[/tex] would veer off to negative infinity. Then
[tex]10^{4x}\to10^{\text{large negative number}}=\dfrac1{10^{\text{large positive number}}}[/tex]
and as [tex]x[/tex] gets larger in magnitude, this would force the expression to converge to 0, so that
[tex]\displaystyle\lim_{x\to-\infty}2\cdot10^{4x}=0[/tex]
