Writing Limits of Behavior: limit of f(x) = sqrt(| x |)

[heartshapes]

Directions: For each of the specific choices of f(x) below, write a very precise and complete description of \(\displaystyle \[ \lim_{x \to b \]\) f(x) for all possible values of real number b. Free free to give separable cases for the various behaviors f(x) can exhibit but be sure you have covered all possibilities for b.

a) f(x)=\(\displaystyle \sqrt{|x|}\)

I have no idea what to do or where to start. My teacher wouldn't give anyone any further help. THANKS!

 

[mmm4444bot]

Re: Writing Limits of Behavior

Hi Heart Shapes:

If your teacher is not providing additional information, then I must assume that you are in a class where students are already supposed to be familiar with (or you have a textbook that explains):

1) function notation

2) the concept of a limit (including notation)

3) the concept of absolute value

4) the behavior of ?x

If you could give us some idea of what you already know, then people at this site would know where to begin to help you.

For example, do you at least understand what the parameter b represents in this exercise?

Please let us know if you are not sure about any of 1, 2, 3, or 4 above, too.

In the meantime, if you do understand function notation (and I think that you probably do), then begin by considering the function g(x) = ?x. Do you know what the graph of this function looks like? What happens to the graph when x is negative? Does putting absolute value symbols around x change anything?

If I asked you what the limit of g(x) is as x approaches 45, then what could you say?

Please let us know.

Cheers,

~ Mark

 

[pka]

Re: Writing Limits of Behavior

The domain of \(\displaystyle f(x)=\sqrt {\left|x\right|}\) is the set of all real numbers.
Therefore, \(\displaystyle \lim _{x \to b}\sqrt {\left|x\right|}=\sqrt{\left|b\right|}\).
Although straighforward, that does not seem to be what is required.
So why not post the other problems? Prehaps there would be a clue as to what is wanted.

 

[heartshapes]

Re: Writing Limits of Behavior

oh sorry. I do know all of a through d.

I know g(x) is only in the first quadrant and it is kind of like half of a sideways u.

Is b in f(x) positive and negative infinity? Because you can plug in any number in for the value of x and get a positive value?

 

[heartshapes]

Re: Writing Limits of Behavior

The domain of \(\displaystyle f(x)=\sqrt {\left|x\right|}\) is the set of all real numbers.
Therefore, \(\displaystyle \lim _{x \to b}\sqrt {\left|x\right|}=\sqrt{\left|b\right|}\).
Although straighforward, that does not seem to be what is required.
So why not post the other problems? Prehaps there would be a clue as to what is wanted.

The other problems are the same thing...

b) f(x) = tan x

c) f(x) = ln (x)

I just hoped if I figured out how to do a. I'd get the general idea and be able to figure out the rest.

 

[pka]

Re: Writing Limits of Behavior

Does your class know about continuous functions? Have you covered that?
For question b, the limit does not exist at any number for which tangent is not defined.

 

[heartshapes]

Re: Writing Limits of Behavior

I think we have... my class actually started last week and the teacher just expects us to know everything.

oh so for b, wherever sec[sup:3130gh5e]2[/sup:3130gh5e]x isn't defined the limit is undefined?

 

[pka]

Re: Writing Limits of Behavior

so for b, wherever sec[sup:3oi0j6ja]2[/sup:3oi0j6ja]x isn't defined the limit is undefined?

No. The tangent function is not defined at \(\displaystyle \pm \frac{\pi }{2},\; \pm \frac{{3\pi }}{2},\; \pm \frac{{5\pi }}{2},\; \cdots\).
The limit will exist at all other real numbers.

 

[heartshapes]

Re: Writing Limits of Behavior

Oh. Ok thanks.