# How do you find #lim (sqrtx-1)/(root3x-1)# as #x->1# using l'Hospital's Rule?

If we then start the L'Hôpital process, we get:

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To find the limit of (sqrt(x) - 1) / (sqrt(3x) - 1) as x approaches 1 using L'Hôpital's Rule, we first notice that if we directly substitute x = 1 into the expression, we get an indeterminate form of 0/0.

Now, applying L'Hôpital's Rule, we differentiate the numerator and the denominator separately with respect to x.

For the numerator: d/dx (sqrt(x) - 1) = (1/2)*(1/sqrt(x))

For the denominator:
d/dx (sqrt(3x) - 1) = (1/2)*(1/sqrt(3x))*(3)

Now, we evaluate these derivatives at x = 1:

Numerator: (1/2)*(1/sqrt(1)) = 1/2
Denominator: (1/2)*(1/sqrt(3)) * 3 = 3/(2*sqrt(3))

So, the limit of the original expression as x approaches 1 is the same as the limit of the derivatives:

lim (x->1) [(sqrt(x) - 1) / (sqrt(3x) - 1)] = lim (x->1) [(1/2) / (3/(2*sqrt(3)))]

Now, we can directly substitute x = 1 into this expression:

lim (x->1) [(1/2) / (3/(2*sqrt(3)))] = (1/2) / (3/(2*sqrt(3))) = sqrt(3)/3

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To find ( \lim_{x \to 1} \frac{\sqrt{x} - 1}{\sqrt{3x} - 1} ) using L'Hospital's Rule:

- Apply L'Hospital's Rule by taking the derivative of the numerator and the derivative of the denominator separately.
- Differentiate the numerator: ( \frac{d}{dx} (\sqrt{x} - 1) = \frac{1}{2\sqrt{x}} ).
- Differentiate the denominator: ( \frac{d}{dx} (\sqrt{3x} - 1) = \frac{1}{2\sqrt{3x}} \times 3 = \frac{3}{2\sqrt{3x}} ).
- Substitute these derivatives into the expression: ( \lim_{x \to 1} \frac{\frac{1}{2\sqrt{x}}}{\frac{3}{2\sqrt{3x}}} ).
- Simplify the expression: ( \lim_{x \to 1} \frac{\sqrt{3x}}{3\sqrt{x}} ).
- Substitute ( x = 1 ) into the simplified expression: ( \frac{\sqrt{3(1)}}{3\sqrt{1}} = \frac{\sqrt{3}}{3} ).

Thus, ( \lim_{x \to 1} \frac{\sqrt{x} - 1}{\sqrt{3x} - 1} = \frac{\sqrt{3}}{3} ).

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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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