Given the function below, determine if the function is continuous at the point
x
=
2
\displaystyle {x}={2}
x
=
2
. If not, indicate why.
f
(
x
)
=
lim
x
→
2
4
−
x
2
2
−
x
\displaystyle {f{{\left({x}\right)}}}=\lim_{{{x}\rightarrow{2}}}\ \frac{{{4}-{x}^{{2}}}}{{{2}-{x}}}
f
(
x
)
=
x
→
2
lim
2
−
x
4
−
x
2
Continuous at
x
=
2
\displaystyle {x}={2}
x
=
2
Not continuous:
f
(
2
)
\displaystyle {f{{\left({2}\right)}}}
f
(
2
)
is not defined; this is a removable discontinuity
Not continuous:
f
(
2
)
\displaystyle {f{{\left({2}\right)}}}
f
(
2
)
is not defined; this is
not
a removable discontinuity
Not continuous:
lim
x
→
2
f
(
x
)
\displaystyle \lim_{{{x}\rightarrow{2}}}{f{{\left({x}\right)}}}
x
→
2
lim
f
(
x
)
does not exist
Not continuous:
f
(
2
)
\displaystyle {f{{\left({2}\right)}}}
f
(
2
)
and limit exist, but are not equal
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