Answer: We can create a table of values to approximate the limit:
x (cos(8x) + 32x^2 - 1) / x^4
-----------------------------------
0.1 2330.04
0.01 233000.04
0.001 23300000.04
0.0001 2329999999.57
0.00001 232999999971.61
As x approaches 0, we can see that the expression (cos(8x) + 32x^2 - 1) / x^4 becomes very large and positive. Therefore, we can say that:
lim (cos(8x) + 32x^2 - 1) / x^4
x->0 = +infinity
In other words, the limit does not exist in the real number system.
Explanation: