Question

let g(x) and h(x) be defined by g(x) = -5x² + 5x + 5 and h(x) = -5x + 10. if f(x) is a function such that g(x) ≤ f(x) ≤ h(x), find \\(\lim\limits_{x \to 1} f(x)\\), or state that it cannot be determined.

Explanation:

Step1: Find limit of \( g(x) \) as \( x \to 1 \)

Substitute \( x = 1 \) into \( g(x)=-5x^{2}+5x + 5 \).
\( \lim_{x\to 1}g(x)=-5(1)^{2}+5(1)+5=-5 + 5+5 = 5 \)

Step2: Find limit of \( h(x) \) as \( x \to 1 \)

Substitute \( x = 1 \) into \( h(x)=-5x + 10 \).
\( \lim_{x\to 1}h(x)=-5(1)+10=-5 + 10 = 5 \)

Step3: Apply Squeeze Theorem

Since \( g(x)\leq f(x)\leq h(x) \) for all \( x \) (in the domain) and \( \lim_{x\to 1}g(x)=\lim_{x\to 1}h(x) = 5 \), by the Squeeze Theorem, \( \lim_{x\to 1}f(x)=5 \).

Answer:

\( 5 \)