# How do you find the limit of # (sqrt(3x - 2) - sqrt(x + 2))/(x-2)# as x approaches 2?

So, without further ado, let's get to it.

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If your calculus course has not yet covered derivatives and l'Hospital's rule, use algebra to rationalize the numerator.

So we have

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To find the limit of the given expression as x approaches 2, we can simplify it using algebraic manipulation. By multiplying the numerator and denominator by the conjugate of the numerator, we can eliminate the square roots.

The conjugate of the numerator is (sqrt(3x - 2) + sqrt(x + 2)).

Multiplying the numerator and denominator by the conjugate, we get:

[(sqrt(3x - 2) - sqrt(x + 2))/(x-2)] * [(sqrt(3x - 2) + sqrt(x + 2))/(sqrt(3x - 2) + sqrt(x + 2))]

Expanding the numerator using the difference of squares, we have:

[(3x - 2) - (x + 2)] / [(x - 2)(sqrt(3x - 2) + sqrt(x + 2))]

Simplifying further, we get:

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

Now, we can cancel out the common factor of (x - 2) in the numerator and denominator:

(2(x - 2)) / [(x - 2)(sqrt(3x - 2) + sqrt(x + 2))]

Simplifying again, we have:

2 / (sqrt(3x - 2) + sqrt(x + 2))

Now, we can substitute x = 2 into the expression:

2 / (sqrt(3(2) - 2) + sqrt(2 + 2))

2 / (sqrt(4) + sqrt(4))

2 / (2 + 2)

2 / 4

1/2

Therefore, the limit of the given expression as x approaches 2 is 1/2.

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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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