If 2x ≤ g(x) ≤ x4 − x2 + 2 for all x, evaluate .
To evaluate: The limit value of the function as x approaches 1.
The inequality is, for all x.
Suppose that c is a constant and the limits and exist. Then
Limit law 1:
Limit law 2:
Limit law 3:
Limit law 7:
Limit law 8:
Limit law 9: where n is a positive integer.
Theorem used: The Squeeze Theorem
“If when x is near a (except possibly at a) and then .”
Apply the Squeeze Theorem and obtain a function f smaller than and a function h bigger than such that both and approaches 4.
The given inequality becomes, .
When the limit x approaches to 1, the inequality becomes,
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