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Mathematics
Mathematics
Problem
Let
k
k
k
be a real number. Define
F
(
x
)
=
x
2
−
k
x
−
2
x
−
1
,
x
≠
1.
F(x)=\frac{x^{2}-kx-2}{x-1},\quad x\neq 1.
F
(
x
)
=
x
−
1
x
2
−
k
x
−
2
,
x
=
1.
(i) Determine the value of
k
k
k
for which
lim
x
→
1
F
(
x
)
\displaystyle \lim_{x\to 1} F(x)
x
→
1
lim
F
(
x
)
exists and equals
3
3
3
. (ii) For this value of
k
k
k
, simplify
F
(
x
)
F(x)
F
(
x
)
and write the continuous extension of
F
F
F
at
x
=
1
x=1
x
=
1
.
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