Final answer:
The limit of x⁰⁾ˣ as x approaches 0+ is 0. This result is achieved by applying L'Hôpital's Rule to the indeterminate form 0⋅(0), which, after differentiation, simplifies to 1/x as x approaches 0+, resulting in the limit being 0.
Step-by-step explanation:
The limit of x⁰⁾ˣ as x approaches 0+ is a classic example of an indeterminate form. To solve this, we can apply L'Hôpital's Rule because as x approaches 0, both x and ln(x) approach 0, resulting in a 0⋅(0) indeterminate form, which suggests that L'Hôpital's Rule might be applicable.
Let's define the function as f(x) = x·ln(x). We differentiate the function's numerator and denominator separately with respect to x, which gives us:
- The derivative of the numerator (ln(x)) is 1/x.
- The derivative of the denominator is 1.
This simplifies our limit problem to the limit of 1/x as x approaches 0+, which is infinity. Consequently, the limit of x⁰⁾ˣ as x approaches 0+ is
0
.