Calculate the following limit...

[flaren5]

lim g(x) given that 1/3 - x^2 ≤ g(x) ≤ 1/3cos x, for all x
(x→0)


...any insight would be greatly appreciated, since I have not taken calculus in quite some time, I need a refresher with limits.

Thank you!!

 

[stapel]

lim g(x) given that 1/3 - x^2 ≤ g(x) ≤ 1/3cos x, for all x
(x→0)

Are the limits either of the following?

. . . . .\(\displaystyle \frac{1}{3}\, -\, x^2\, \mbox{ and }\, \frac{1}{3}\cos(x)\)

. . . . .\(\displaystyle \frac{1}{3\, -\, x^3}\, \mbox{ and }\, \frac{1}{3\cos(x)}\)

I have not taken calculus in quite some time, I need a refresher with limits.

The subject of limits is usually covered in at least one chapter of a calculus text. There is no "fifty words or less" "refresher". Are you saying that you don't remember anything at all, so you're needing lesson instruction on this topic, or that you're not sure of what you're done, and would appreciate a check of your work so far? If the latter, please reply with what you've tried. Thank you! :wink:

 

[stapel]

Are the limits either of the following?

. . . . .\(\displaystyle \frac{1}{3}\, -\, x^2\, \mbox{ and }\, \frac{1}{3}\cos(x)\)

. . . . .\(\displaystyle \frac{1}{3\, -\, x^3}\, \mbox{ and }\, \frac{1}{3\cos(x)}\)

Until LaTeX formatting is functional again, let me rephrase:

(1/3) - x^2 and (1/3)cos(x)
or
1/(3 - x^2) and 1/(3cos(x))

 

[flaren5]

Until LaTeX formatting is functional again, let me rephrase:

(1/3) - x^2 and (1/3)cos(x)
or
1/(3 - x^2) and 1/(3cos(x))[/SIZE][/SIZE]

The following is what is being asked,

(1/3) - x^2 and (1/3)cos(x)

I have read the chapter, I`ll have to go over it again and again...I seem to be understanding some of the concepts, I think with this particular problem I may be getting hung up on the cos(x). Thank you for responding, any insight of how I would go about this would be appreciated.

 

[Deleted member 4993]

lim g(x) given that 1/3 - x^2 ≤ g(x) ≤ 1/3cos x, for all x
(x→0)


...any insight would be greatly appreciated, since I have not taken calculus in quite some time, I need a refresher with limits.

Thank you!!

Lim(x→0)[1/3 - x²] = ??

Lim(x→0)[1/3 * cos(x)] = ??

Since g(x) is squeezed between these two functions - what would be the limit of g(x) as x→0?

For limit problems, I find it extremely useful to use graphing calculator and stare at the graph to comprehend the concept of the problem.

 

[DrPhil]

The following is what is being asked,

(1/3) - x^2 and (1/3)cos(x)

I have read the chapter, I`ll have to go over it again and again...I seem to be understanding some of the concepts, I think with this particular problem I may be getting hung up on the cos(x). Thank you for responding, any insight of how I would go about this would be appreciated.

As x approaches 0, you can replace cosine by the first terms of its power-series expansion:

\(\displaystyle \cos{x} = 1 - \frac{1}{2} x^2 + \cdot \cdot \cdot \)

\(\displaystyle \frac{1}{3} - x^2 \le g(x) \le \frac{1}{3} -\frac{1}{6} x^2 \)