How do you find #lim (sqrt(9x)3)/x# as #x>0# using l'Hospital's Rule?
To find the limit (\lim_{x \to 0} \frac{\sqrt{9x}  3}{x}) using L'Hôpital's Rule, follow these steps:

Check if the limit is in an indeterminate form ( \frac{0}{0} ) or ( \frac{\infty}{\infty} ).

If it's in an indeterminate form, differentiate the numerator and the denominator separately.

Evaluate the limit of the resulting expression as ( x ) approaches the limit point.
Here's how to apply L'Hôpital's Rule to this problem:

Check the form of the limit: [ \lim_{x \to 0} \frac{\sqrt{9x}  3}{x} ] At ( x = 0 ), we have ( \frac{0}{0} ), indicating an indeterminate form.

Apply L'Hôpital's Rule: [ \lim_{x \to 0} \frac{\frac{d}{dx}(\sqrt{9x}  3)}{\frac{d}{dx}(x)} ]

Differentiate the numerator and the denominator: [ \lim_{x \to 0} \frac{\frac{d}{dx}(\sqrt{9x}  3)}{\frac{d}{dx}(x)} = \lim_{x \to 0} \frac{\frac{d}{dx}(\sqrt{9x})}{1} ] We differentiate ( \sqrt{9x} ) using the chain rule: [ = \lim_{x \to 0} \frac{\frac{1}{2\sqrt{9x}} \cdot (1)}{1} ]

Simplify and evaluate the limit: [ = \lim_{x \to 0} \frac{1}{2\sqrt{9x}} = \frac{1}{2\sqrt{90}} = \frac{1}{6} ]
So, ( \lim_{x \to 0} \frac{\sqrt{9x}  3}{x} = \frac{1}{6} ).
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The answer is
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This is undefined, so we apply L'Hôspital's Rule
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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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