Evaluate the limit of the indeterminate quotient?
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Evaluate the limit of the indeterminate quotient?
┏─━─━─━─━∞◆∞━─━─━─━─┓
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┗─━─━─━─━∞◆∞━─━─━─━─┛
Answer:
lim
x
→
0
√
7
−
x
−
√
7
+
x
x
=
−
1
√
7
Explanation:
Evaluate the limit:
lim
x
→
0
√
7
−
x
−
√
7
+
x
x
Rationalize the numerator of the function using the identity:
(
a
+
b
)
(
a
−
b
)
=
(
a
2
−
b
2
)
:
√
7
−
x
−
√
7
+
x
x
=
(
√
7
−
x
−
√
7
+
x
x
)
(
√
7
−
x
+
√
7
+
x
√
7
−
x
+
√
7
+
x
)
=
7
−
x
−
7
−
x
x
(
√
7
−
x
+
√
7
+
x
)
=
−
2
x
x
(
√
7
−
x
+
√
7
+
x
)
=
−
2
√
7
−
x
+
√
7
+
x
Now the limit is determinate:
lim
x
→
0
√
7
−
x
−
√
7
+
x
x
=
lim
x
→
0
−
2
√
7
−
x
+
√
7
+
x
=
−
1
√
7
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