Question

PLEASE HELP WILL GIVE BRAINLESS and 100 points

Answer 1

D) Cannot be determined based on the information.

Explanation:

The given functions g(x) and f(x) are:

`f(x) = x^2 - 2x + 4`

`g(x)=(1)/(3)x^2+(2)/(5)`

According to the given conditions, h(x) ≥ g(x) and h(x) ≤ f(x) for all values of x. This means that h(x) is bounded between g(x) and f(x) for all x.

Therefore:

`g(x) \leq h(x) \leq f(x)`

`(1)/(3)x^2 + (2)/(5) \leq h(x) \leq x^2 - 2x + 4`

Consider the limit as x approaches 0 for g(x) and f(x):

`\displaystyle \lim_(x \to 0) (g(x))=\lim_(x \to 0) \left((1)/(3)x^2 + (2)/(5)\right) = (2)/(5)`

`\displaystyle \lim_(x \to 0)(f(x))=\lim_(x \to 0) \left(x^2 - 2x + 4\right) = 4`

`\boxed{\begin{array}{c}\underline{\sf Squeeze\;Theorem}\\\\\textsf{If}\;g(x)\leq h(x)\leq f(x)\;\textsf{when $x$ is near $a$, except possibly at $a$,}\\\\\textsf{and}\; \displaystyle \lim_(x \to a)f(x)=\lim_(x \to a)g(x)=L,\; \textsf{then}\; \lim_(x \to a) h(x)=L.\end{array}}`

The Squeeze Theorem states that if h(x) is squeezed between f(x) and g(x) near
`a`, and if f(x) and g(x) have the same limit L at
`a`, then h(x) is trapped and will be forced to have the same limit L at
`a` also.

As the limits of the two bounding functions are different, we cannot determine the limit of h(x) as x approaches 0 based solely on the information given.