lim
β‘
π₯
β
π
1
β
2
π₯
π₯
2
β
4
.
If this is the intended form, let's analyze it:
Step-by-Step Explanation:
Denominator Simplification:
The denominator
π₯
2
β
4
is a difference of squares, which can be factored as:
π₯
2
β
4
=
(
π₯
β
2
)
(
π₯
+
2
)
.
Numerator Behavior:
The numerator is
1
β
2
π₯
. As
π₯
β
π
, the behavior depends on the specific value of
π
. For example:
If
π₯
β
0
, the term
2
π₯
becomes undefined (division by zero).
For other values of
π₯
, the numerator takes finite values.
Evaluating the Limit:
To fully analyze and evaluate the limit, the value of
π
(where
π₯
approaches) must be specified. In particular:
If
π₯
β
2
, both the numerator and denominator might lead to indeterminate forms (e.g.,
0
0
).
In such cases, techniques like LβHΓ΄pitalβs Rule or algebraic simplification can help resolve the limit
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