Steven G. answered • 10/22/19

College Math Professor

Not that if f(x) is any polynomial, then lim as x goes to a of f(x) is just f(a). That is it is just evaluation of f(x) for x=a (or more simply, the lim is f(a)

Let g and h be the functions defined by g(x)=−2x^2+4x+1 and h(x)=(1/2)x^2−x+(11/2). If f is a function that satisfies g(x) ≤ f(x) ≤ h(x) for all x, what is lim f(x) as x→1?

What is the limit of g(x) as x goes to 1? What is the limit of h(x) as x goes to 1?

If you get the same answer than lim f(x) as x→1 will be that common answer.

If you do not get a common answer than you can NOT determine the lim f(x) as x→1

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Let f, g, and h be the functions defined by f(x)= (1−cosx)/x^2, g(x)= x^2(sin(1/x)), and h(x)= (sinx)/x for x≠0. All of the following inequalities are true for x≠0. Which of the inequalities can be used with the squeeze theorem to find the limit of the function as x approaches 0 ?

1- (1/3)(1−x^2) ≤ f(x) ≤ (1/2)

2- −x^2 ≤ g(x) ≤ x^2

3- −(1/ |x|) ≤ h(x) ≤ (1/ |x|)

Unless I am missing something, you do not have to be given what f, g and h are.

In each inequality, determine the lim as x approaches 0 for the lhs and the rhs of the inequality. If you get a common answer then yes you can use the squeeze theorem, otherwise you can't