Question

guess the value of the limit lim x→[infinity] x6 6x by evaluating the function f(x) = x6 6x for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. use a graph of f to support your guess

Answer 1

The limit of the function f(x) = x^6 - 6x as x approaches infinity is infinity. This is because the x^6 term dominates, growing much faster than the 6x term, which is demonstrated by evaluating f(x) at various values and confirmed by its graph.

Step-by-step explanation:

The student is asking to evaluate the limit of the function f(x) = x^6 - 6x as x approaches infinity. When we look at the given function, as x gets larger, the x^6 term will grow much faster than the linear 6x term. This means the behavior of f(x) will be dominated by x^6 as x grows larger. Therefore, we can expect that the value of the function will approach infinity as well.

To confirm this, we can evaluate the function at various values of x and observe the trend. The function value increases as x increases and becomes quite large, even for relatively small values of x. A graph would show that the curve heads upwards sharply as x increases. This observation supports the claim that the limit of f(x) as x approaches infinity is indeed infinity.