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Mathematics
Mathematics
Problem
For
k
>
0
k>0
k
>
0
, determine the value of
k
k
k
that satisfies
lim
x
→
1
k
x
+
1
−
k
+
1
x
−
1
=
1
3
\displaystyle \lim_{x \to 1} \frac{\sqrt{kx+1}-\sqrt{k+1}}{x-1}=\dfrac{1}{3}
x
→
1
lim
x
−
1
k
x
+
1
−
k
+
1
=
3
1
. a)
2
(
1
+
13
)
9
\dfrac{2\bigl(1+\sqrt{13}\bigr)}{9}
9
2
(
1
+
13
)
b)
2
(
1
+
10
)
9
\dfrac{2\bigl(1+\sqrt{10}\bigr)}{9}
9
2
(
1
+
10
)
c)
1
+
10
9
\dfrac{1+\sqrt{10}}{9}
9
1
+
10
d)
5
4
\dfrac{5}{4}
4
5
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