Evaluate limit L'Hospitals Rule:
lim x->0+ (x)^(pi/2)
To evaluate the limit using L'Hospital's Rule, we need to take the derivative of both the numerator and the denominator until we obtain an indeterminate form, such as 0/0 or ∞/∞. Let's start by finding the derivative of the numerator and the denominator.
The numerator is x^(pi/2), so let's take its derivative:
(d/dx) (x^(pi/2)) = (pi/2) * (x^(pi/2 - 1)) = (pi/2) * (x^(-1 + pi/2)) = (pi/2) * (x^(-1/2)).
The denominator is x, so its derivative is:
(d/dx) (x) = 1.
Now, we have the derivative of both the numerator and the denominator. Next, we will evaluate the limit of the derivatives as x approaches 0+.
lim x->0+ [(pi/2) * (x^(-1/2))] / 1 = (pi/2) * lim x->0+ (x^(-1/2)).
Continuing with L'Hospital's Rule, we can take the derivative of (x^(-1/2)):
(d/dx) (x^(-1/2)) = (-1/2) * (x^(-1/2 - 1)) = (-1/2) * (x^(-3/2)).
Now, we can evaluate the limit of the new expression:
(pi/2) * lim x->0+ [(-1/2) * (x^(-3/2))] = (pi/2) * [(-1/2) * (0^(-3/2))].
At this point, we have an indeterminate form of (-∞) * (0^(-3/2)). To simplify this, we can rewrite 0^(-3/2) as 0.
Thus, the final evaluation of the limit using L'Hospital's Rule is:
lim x->0+ (x)^(pi/2) = (pi/2) * [(-1/2) * 0] = 0.