Question

What is the limit as x approaches infinity of x to the power of x?

a) 0
b) 1
c) [infinity]
d) undefined

Answer 1

The limit of x^x as x approaches infinity is undefined because the value grows too rapidly, not approaching a finite number or becoming steady.

Step-by-step explanation:

The Limit of x to the Power of x as x Approaches Infinity

The function f(x) = x^x is an interesting case when it comes to finding limits as x approaches infinity. Intuitively, both the base x and the exponent x increase without bound, leading the function to grow at an increasingly rapid rate. Therefore, the limit of xx as x approaches infinity is:

d. Undefined

As x becomes very large, the function x^x will grow faster than basic exponential functions like 2^x or e^x because the base itself is also growing. Thus, it does not approach a finite number nor does it level off to a steady value, making the limit undefined. Other functions, such as y = 1/x, display asymptotic behavior where they approach a finite limit, but in the case of x^x, it keeps growing without bound.