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## Main Question or Discussion Point

Hi all,

My girlfriend asked me this question just now, however I have no idea how I should approach to solve it, I highly appreciate if anyone could shed lights on this:

lim x^2 times cos^2(x^-2)

x->0

I tried using the squeeze theorem:

-1 < cos(1/x) < 1

thus:

-x^2 < x^2 (cos(1/x)) < x^2

or

-x^2 cos(1/x) < x^2 (cos^2(x^-2)) < x^2 cos(1/x)

Therefore, as x->0 in the middle, the two sides also approach 0.

But I don't think it makes any sense... since 1/x as x-> 0 cannot really be used as part of the intervals representing -1 and 1.

I also tried rearranging cos^2 (x^-2), but I don't think it's any use.

Please enlighten on this, thanks in advance.

My girlfriend asked me this question just now, however I have no idea how I should approach to solve it, I highly appreciate if anyone could shed lights on this:

lim x^2 times cos^2(x^-2)

x->0

I tried using the squeeze theorem:

-1 < cos(1/x) < 1

thus:

-x^2 < x^2 (cos(1/x)) < x^2

or

-x^2 cos(1/x) < x^2 (cos^2(x^-2)) < x^2 cos(1/x)

Therefore, as x->0 in the middle, the two sides also approach 0.

But I don't think it makes any sense... since 1/x as x-> 0 cannot really be used as part of the intervals representing -1 and 1.

I also tried rearranging cos^2 (x^-2), but I don't think it's any use.

Please enlighten on this, thanks in advance.

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