# How do you find the limit of #(2 - root3x) / (sqrt(x - 4) - 2)# as x approaches 8?

Use the conjugates of the numerator and the denominator to find that the limit is

Consider, then, third powers and third roots.

I hope that you learned in your previous study of algebra that

On to the question at hand:

We can now evaluate the limit by substitution. The form is no longer indeterminate.

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To find the limit of (2 - √(3x)) / (√(x - 4) - 2) as x approaches 8, we can use algebraic manipulation and the concept of conjugate pairs.

First, we notice that both the numerator and denominator involve square roots. To eliminate the square roots, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (√(x - 4) + 2).

By multiplying the numerator and denominator by (√(x - 4) + 2), we get:

[(2 - √(3x)) * (√(x - 4) + 2)] / [(√(x - 4) - 2) * (√(x - 4) + 2)]

Expanding and simplifying this expression, we have:

[(2√(x - 4) - √(3x)(√(x - 4)) + 4 - 2√(3x))] / [(x - 4) - 4]

Further simplifying, we get:

[(2√(x - 4) - √(3x)(√(x - 4)) + 4 - 2√(3x))] / (x - 8)

Now, we can cancel out the common terms in the numerator:

[2√(x - 4) - √(3x)(√(x - 4)) - 2√(3x)] / (x - 8)

As x approaches 8, we substitute this value into the expression:

[2√(8 - 4) - √(3 * 8)(√(8 - 4)) - 2√(3 * 8)] / (8 - 8)

Simplifying further:

[2√4 - √24 - 2√24] / 0

Since the denominator is 0, we cannot directly evaluate the limit using this method. We need to try a different approach, such as L'Hôpital's Rule or factoring and simplifying the expression further.

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To find the limit of the given expression as ( x ) approaches 8, you can rationalize the expression by multiplying both the numerator and the denominator by the conjugate of the denominator. This will help eliminate the radical in the denominator. After rationalizing, you can then substitute ( x = 8 ) into the expression and evaluate the limit.

[ \lim_{x \to 8} \frac{2 - \sqrt{3x}}{\sqrt{x - 4} - 2} ]

Rationalizing the expression:

[ \lim_{x \to 8} \frac{(2 - \sqrt{3x})(\sqrt{x - 4} + 2)}{(\sqrt{x - 4} - 2)(\sqrt{x - 4} + 2)} ]

[ = \lim_{x \to 8} \frac{(2 - \sqrt{3x})(\sqrt{x - 4} + 2)}{x - 4 - 4} ]

[ = \lim_{x \to 8} \frac{(2 - \sqrt{3x})(\sqrt{x - 4} + 2)}{x - 8} ]

[ = \lim_{x \to 8} \frac{(2 - \sqrt{3x})(\sqrt{x - 4} + 2)}{x - 8} ]

[ = \frac{(2 - \sqrt{3 \cdot 8})(\sqrt{8 - 4} + 2)}{8 - 8} ]

[ = \frac{(2 - \sqrt{24})(\sqrt{4} + 2)}{0} ]

[ = \frac{(2 - 2\sqrt{6})(2 + 2)}{0} ]

[ = \frac{4 - 4\sqrt{6}}{0} ]

As ( x ) approaches 8, the denominator approaches 0. Hence, the limit is undefined or diverges.

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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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