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

Answer 1

#= 3/2#

#lim_(x to 1) (sqrtx-1)/(root3x-1)#
If we plug in #x = 1#, we obtain #0/0#, so this is in indeterminate form and we can use L'Hôpital's Rule.

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

#= lim_(x to 1) (1/2 x^(- 1/2))/(1/3 x^(- 2/3))#
At this point we can try #x = 1# again, and we get:
#= (1/2 (1)^(-1/2))/(1/3 (1)^(- 2/3)) = 3/2#
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Answer 2

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/(2sqrt(3)))] = (1/2) / (3/(2sqrt(3))) = sqrt(3)/3

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Answer 3

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

  1. Apply L'Hospital's Rule by taking the derivative of the numerator and the derivative of the denominator separately.
  2. Differentiate the numerator: ( \frac{d}{dx} (\sqrt{x} - 1) = \frac{1}{2\sqrt{x}} ).
  3. Differentiate the denominator: ( \frac{d}{dx} (\sqrt{3x} - 1) = \frac{1}{2\sqrt{3x}} \times 3 = \frac{3}{2\sqrt{3x}} ).
  4. Substitute these derivatives into the expression: ( \lim_{x \to 1} \frac{\frac{1}{2\sqrt{x}}}{\frac{3}{2\sqrt{3x}}} ).
  5. Simplify the expression: ( \lim_{x \to 1} \frac{\sqrt{3x}}{3\sqrt{x}} ).
  6. 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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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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