How do you find #lim (sqrt(9-x)-3)/x# as #x->0# using l'Hospital's Rule?

Answer 1

To find the limit (\lim_{x \to 0} \frac{\sqrt{9-x} - 3}{x}) using L'Hôpital's Rule, follow these steps:

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

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

  3. 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:

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

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

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

  4. Simplify and evaluate the limit: [ = \lim_{x \to 0} \frac{-1}{2\sqrt{9-x}} = \frac{-1}{2\sqrt{9-0}} = -\frac{1}{6} ]

So, ( \lim_{x \to 0} \frac{\sqrt{9-x} - 3}{x} = -\frac{1}{6} ).

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Answer 2

The answer is #=-1/6#

We need

#(sqrtx)'=1/(2sqrtx)#
#lim_(x->0)(sqrt(9-x)-3)/x=0/0#

This is undefined, so we apply L'Hôspital's Rule

#lim_(x->0)(sqrt(9-x)-3)/x=lim_(x->0)((sqrt(9-x)-3)')/(x')#
#=lim_(x->0)(-1/(2sqrt(9-x)-3))#
#=-1/6#
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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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