Find the domain and range of this function:

f(x) = cosx/x.

I just took a quiz today and I blanked out. I wasn't sure what notation you are supposed to use for this kind of problem so I probably got it wrong.

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- Sep 2nd 2011, 05:40 PMexplodingtoenailsDomain and range of this function?
Find the domain and range of this function:

f(x) = cosx/x.

I just took a quiz today and I blanked out. I wasn't sure what notation you are supposed to use for this kind of problem so I probably got it wrong. - Sep 2nd 2011, 07:20 PMProve ItRe: Domain and range of this function?
- Sep 2nd 2011, 08:10 PMexplodingtoenailsRe: Domain and range of this function?
The domain would be (-infinity, 0) U (0, infinity)?

- Sep 2nd 2011, 08:17 PMProve ItRe: Domain and range of this function?
Correct, though it's easier to write $\displaystyle \displaystyle \mathbf{R}\backslash\{0\}$

Now the range is a bit tougher, but not impossible. You should know that the cosine function oscillates between -1 and 1. What happens when the denominator is large? When the denominator is small? - Sep 2nd 2011, 08:56 PMexplodingtoenailsRe: Domain and range of this function?
When the denominator is large, the number is smaller, and vice versa when the denominator is small.

Would the range be (-infinity, +infinity)?

Also, how do I know which notation to use when doing domain and range of a function? ie. The parenthetical notation or using "less than or equal to" signs? - Sep 2nd 2011, 09:36 PMProve ItRe: Domain and range of this function?
Yes, because for small negative values of x, the function shoots to $\displaystyle \displaystyle -\infty$, while for small positive values of $\displaystyle \displaystyle x$ the function shoots to $\displaystyle \displaystyle +\infty$. Of course, it's easier to write $\displaystyle \displaystyle \mathbf{R}$ instead of $\displaystyle \displaystyle (-\infty, \infty)$.

All notations are equivalent, it's just personal preference (I go for whichever one is easiest to write in each case). - Sep 3rd 2011, 12:41 PMexplodingtoenailsRe: Domain and range of this function?
Just as an extra question, would the limit as x approaches infinity of cosx/x be infinity?

- Sep 3rd 2011, 12:52 PMProve ItRe: Domain and range of this function?
- Sep 3rd 2011, 01:08 PMexplodingtoenailsRe: Domain and range of this function?
- Sep 3rd 2011, 01:12 PMProve ItRe: Domain and range of this function?
I suggest you zoom out so that you get a better picture of what the function's behaviour is. It fluctuates because the cosine function fluctuates. But you should notice that they fluctuate less and less as the x values increase. Even though the function will continue to oscillate forever, as the denominator increases, the function will still get infinitesimally small so that the function is indistinguishable from 0.

- Sep 3rd 2011, 01:20 PMexplodingtoenailsRe: Domain and range of this function?