Ali, a calculus student, writes the following explanation for why the limit lim𝑥→0[𝑥^4⋅cos(2/3𝑥)] is zero

Question

Ali, a calculus student, writes the following explanation for why the limit lim𝑥→0[𝑥⁴⋅cos(2/3𝑥)] is zero:

Since lim𝑥→0 𝑥⁴=0 we conclude, from the basic limit laws, that

lim𝑥→0[𝑥⁴⋅cos(2/3𝑥)]=(lim𝑥→0x⁴)•[lim𝑥→0cos(2/3𝑥)]=0•[lim𝑥→0cos(2/3𝑥)]=0

What is wrong with Ali's explanation? How would you correct it? 

Answer 1

I think the problem with Ali's explanation has to do with the behavior of the cosine function as it's argument becomes infinitely large.

As x→0 2/(3x)→∞

(actually +∞ approaching from the right and –∞ approaching from the left, but this doesn't matter because cosine is an even function)

As x→0 and 2/(3x) gets infinitely large in magnitude, the value of cos (2/(3x)) fluctuates wildly

So lim_𝑥→0 [cos (2/(3x))] is undefined, which means

lim_𝑥→0 x⁴[cos (2/(3x))] is also undefined

Answer 2

There is no limit L=cos(2/(3x)) x->0

-1≤cos(2/(3x))≤1 when x->0,,,,cos(2/(3x)) never approaches +/- infinity

0*a=0 if -1≤a≤1

therefore.....limit is zero.