Final answer:
To show the limit lim(x→0) x⁸ sin(1/x) is true, we can use the squeeze theorem and find two other functions that the original function is sandwiched between. By applying the squeeze theorem, we can conclude that the limit of x⁸ sin(1/x) as x approaches 0 is 0.
Step-by-step explanation:
To show that the limit limx → 0 x⁸ sin(1/x) is true, we can use the squeeze theorem. Let's consider the boundaries of the function and find two other functions that the original function is sandwiched between.
Since |sin(1/x)| is always less than or equal to 1 for any x, we have -x⁸ ≤ x⁸ sin(1/x) ≤ x⁸. Now, as x approaches 0, both -x⁸ and x⁸ approach 0. Therefore, by the squeeze theorem, the limit of x⁸ sin(1/x) as x approaches 0 is also 0.