Question

Let f, g, and h satisfy f(x) ≤ g(x) ≤ h(x) for all x in some common domain A. If lim ₓ → c f(x)=L and lim ₓ → c h(x)=L at some limit point c of A, show lim ₓ → c g(x)=L as well.

Answer 1

Using the Squeeze Theorem, one can show that if f(x) and h(x) approach the same limit L as x approaches c, g(x) must also approach L since it is bounded by f(x) and h(x).

Step-by-step explanation:

The student's question revolves around limit properties in calculus, asking to show that if for all x in some domain A, we have f(x) ≤ g(x) ≤ h(x) and lim ₓ → c f(x)=L and lim ₓ → c h(x)=L, then it must also be true that lim ₓ → c g(x)=L. To demonstrate this, we can use the Squeeze Theorem, which states that if f(x) and h(x) converge to the same limit L as x approaches c, and g(x) is always between f(x) and h(x), then g(x) must also converge to L as x approaches c.

To apply this theorem, consider that as x approaches c, f(x) approaches L from below or equal to L and h(x) approaches L from above or equal to L. Since g(x) is squeezed in between f(x) and h(x), it cannot but approach L itself, because there is no room for it to converge to any other limit without violating the given inequalities f(x) ≤ g(x) ≤ h(x).

Therefore, by the Squeeze Theorem, we have established that g(x) must approach the same limit L that f(x) and h(x) do, confirming lim ₓ → c g(x)=L.