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.

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

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

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

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

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

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 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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