Evaluate lim→0
2
2−33
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f(x) = ( √
9+x2−3
x2 x 6= 0
a x = 0
continuous everywhere?
Answer:
1
6
Solution: For f(x) to be continuous at x = 0, we must choose a so that:
lim
x→0
f(x) = lim
x→0
√
9 + x
2 − 3
x
2
= f(0) = a.
So we have
a = lim
x→0
√
9 + x
2 − 3
x
2
= lim
x→0
(
√
9 + x
2 − 3)
x
2
(
√
9 + x
2 + 3)
(
√
9 + x
2 + 3)
= lim
x→0
9 + x
2 − 9
x
2(
√
9 + x
2 + 3)
=
lim
x→0
x
2
x
2(
√
9 + x
2 + 3)
= lim
x→0
1
(
√
9 + x
2 + 3)
=
1
√
9 + 3
=
1
6
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