How do you use L'hospital's rule to find the limit #lim_(x->oo)ln(x)/sqrt(x)# ?

To use L'Hôpital's Rule to find the limit of ( \lim_{x \to \infty} \frac{\ln(x)}{\sqrt{x}} ), we first rewrite the expression in an indeterminate form, which is ( \frac{\infty}{\infty} ) in this case. Then, we differentiate the numerator and the denominator separately with respect to (x) until we no longer have an indeterminate form.

1. Rewrite the expression: ( \lim_{x \to \infty} \frac{\ln(x)}{\sqrt{x}} )

2. Differentiate the numerator: ( \lim_{x \to \infty} \frac{1/x}{\sqrt{x}} )

3. Differentiate the denominator: ( \lim_{x \to \infty} \frac{-1/2\sqrt{x}}{1} )

4. Simplify: ( \lim_{x \to \infty} \frac{1}{x\sqrt{x}} )

5. Rewrite the expression: ( \lim_{x \to \infty} \frac{1}{x^{3/2}} )

6. Evaluate the limit as ( x \to \infty ): ( \lim_{x \to \infty} \frac{1}{x^{3/2}} = 0 )

Therefore, ( \lim_{x \to \infty} \frac{\ln(x)}{\sqrt{x}} = 0 ).