Question

If f(x)≤g(x)≤h(x) for all x in a common domain A, and limx→c f(x)=L and lim x→c h(x)=L at some limit point c. What can be concluded about im x --> c g(x)?

Answer 1

Using the Squeeze Theorem, we can conclude that lim x→c g(x) must also be L since g(x) is bounded by f(x) and h(x) which both have a limit of L at c.

Step-by-step explanation:

If we have the inequalities f(x) ≤ g(x) ≤ h(x) for all x in a common domain A, and it is known that limx→c f(x) = L and limx→c h(x) = L for some limit point c, we can conclude something about limx→c g(x) by using the Squeeze Theorem (also known as the Sandwich Theorem).

The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in the domain and limx→c f(x) = limx→c h(x) = L, then limx→c g(x) must also equal L. This is because g(x) is 'squeezed' between f(x) and h(x), and since both f(x) and h(x) approach L as x approaches c, g(x) is forced to approach L as well.