Final answer:
The limit of 64x^2 x 8x needs clarification on the operation between terms. If it is multiplication, the limit of 512x^3 as x approaches any real number is obtained by direct substitution, resulting in the value of the polynomial at that point.
Step-by-step explanation:
The question asks us to find the limit of the expression 64x^2 x 8x. However, there seems to be a typo or missing symbol between 64x^2 and 8x that should indicate the operation to be performed (like addition, subtraction, multiplication, or division). Assuming the missing symbol is a multiplication sign, the correct expression would be 64x^2 * 8x, which simplifies to 512x^3.
To find the limit of 512x^3 as x approaches any real number, you simply substitute the x value into the expression because the function is a polynomial, and polynomials are continuous everywhere. Therefore, the limit as x approaches any real number value is just the value of the polynomial at that point.
If x approaches infinity, the limit of 512x^3 also approaches infinity. Similarly, if x approaches negative infinity, the limit of 512x^3 approaches negative infinity, due to the power of three which keeps the sign of x.
Example:
Find the limit of 512x^3 as x approaches 2:
Limits of polynomials are found by direct substitution:
lim (512x^3) as x -> 2
= 512(2)^3
= 512(8)
= 4096
Thus, the limit as x approaches 2 is 4096.