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Mathematics
Problem
The rational function
g
k
(
x
)
=
k
x
3
−
4
x
+
1
3
x
3
+
5
,
k
∈
R
,
g_k(x)=\frac{k x^{3}-4x+1}{3x^{3}+5}, \qquad k\in\mathbb R,
g
k
(
x
)
=
3
x
3
+
5
k
x
3
−
4
x
+
1
,
k
∈
R
,
has a horizontal asymptote as
x
→
∞
x\to\infty
x
→
∞
. a. Determine the value of
k
k
k
for which
lim
x
→
∞
g
k
(
x
)
=
4
\displaystyle\lim_{x\to\infty}g_k(x)=4
x
→
∞
lim
g
k
(
x
)
=
4
. Show all work leading to your result.
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