Question

limit of f(x) = cos 5 theta/theta as x approaches 0?

Answer 1

Is this (cos(5 theta) )/ theta?

Answer 2

and does theta = x?

Answer 3

All you could do is check the left-sided and right-sided limits. Because otherwise the answer is undefined. Check the left and right and see if both produce the infinity, both produce negative infinity, or they dont match. If they match, then the limit is whatever they match out to be, if they dont match then the limit doesnt exist.

Answer 4

elementalmagicks:yes

Answer 5

Oh, theta squared on bottom. Well still, check the left and check the right and see if they approach the same value of negative infinity or infinity.

Answer 6

Actually if you look at the pdf it asks for the limit approached from the left

Answer 7

they approach negative and positive infinity, so they DNE I beleive, but how do I proove this?

Answer 8

Oh, im not paying attention. Ill back out then, if Im not paying attention to see that, then I shouldnt be answering.

Answer 9

This graph might be helpful to you, if you just want to see the answer: http://www.wolframalpha.com/input/?i=lim+x+goes+to+0-+of+cos%285x%29+%2F+x

Answer 10

thanks that helped :) but how can I prove this problem? do you think graphing it is enough?

Answer 11

Yea you can prove it easily as well. cos(5x) is bounded above and below, while x approaches 0 from the left. Do you see how to show it now?

Answer 12

Oh, there is one important thing you need, other than the boundedness sorry!

Answer 13

You also need the fact that the sign of the numerator is positive.

Answer 14

why does the numerator have to be positive? dont they cancel out?

Answer 15

Yea if they cancel out sometimes and not others it's a problem. Thankfully once you get close enough to x=0, the numerator is always lower bounded and positive. In fact it stays near 1 (since cos 0 = 1).

Answer 16

Thus close to 0 cos(5 x) / x is approximately 1/x

Answer 17

To be more formal and careful we can say that once you get close enough to 0 from the left, we know that 0.5 / x < cos(5 x) < 1/x, and since 0.5 / x -> negative infinity, and so does 1/x, we use the squeeze theorem (or at least the idea from that?), and get the desired result of negative infinity.