AP CALC <span title="LIMITS!?!?!?!?!?!?!?!?!?!?!?">LIMITS!?!?!?!?!?!?!?!?!?!...</span>

Question

i got them all but one, and i can't get it for the life of me

Let (x) = (cos x) / (x)

a Find the domain and range of f

b. Is f even, odd, or neither? Justify your answer.

c. Find the limit as x approaches infinity of f(x)

d. Use the sandwich theorem to justify your answer to part c

it specifically says show all work and DO NOT use a graphing calculator

thanks a million for your answer

Answer 1

a.

-∞≤f(x)≤∞ The values of f(x) as x approaches zero from either positive or negative.

The domain is all numbers excluding 0. You could include imaginary numbers, not sure if you want to do that in this case however.

b.

An even function multiplied by an odd function produces an odd function. Thus the even function cos(x) * the odd function 1/x = the odd function cos(x) / x

c.

The functions limit as x approaches infinity (both negative and positive in this case is zero.

d.

We know that the range of cos(x) for all x (specifically for 0&lt;x&lt;∞) is -1 to 1.

-1 ≤ cos(x) ≤ 1

-1/x ≤ cos(x)/x ≤ 1/x

And knowing the limit of both -1/x and 1/x as x approaches infinity it must follow using the squeeze (or w/e you may call it) thm. that cos(x)/x as x approaches infinity is zero.

Q.E.D.

Please note I took out a few steps, which we&#039;re not needed for the last inequality. Please note if you want to cover negative values of x just reverse the inequality signs. This can thus be used to prove the limit of the function as x approaches negative infinity.