# For all #x>=0# and #2x<=g(x)<=x^4-x^2+2# how do you find the limit of g(x) as #x->1#?

# lim_(x rarr 1) g(x) =2 #

We can use the squeeze theorem (or the sandwich theorem), which basically states that, If:

Then:

So for this problem we have:

And so we can aply the squeeze theorem; which gives

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To find the limit of g(x) as x approaches 1, we need to evaluate the function g(x) at x = 1. By substituting x = 1 into the given inequality, we have 2(1) ≤ g(1) ≤ 1^4 - 1^2 + 2. Simplifying this, we get 2 ≤ g(1) ≤ 2. Therefore, the limit of g(x) as x approaches 1 is 2.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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