# How do you find the Limit of #sqrt (x -1) / (sqrt(x+3) - 2) # as x approaches 1?

Rationalizing the denominator

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To find the limit of sqrt(x - 1) / (sqrt(x + 3) - 2) as x approaches 1, we can use algebraic manipulation.

First, we need to simplify the expression. To do this, we can multiply both the numerator and denominator by the conjugate of the denominator, which is sqrt(x + 3) + 2.

After simplifying, we get (sqrt(x - 1) * (sqrt(x + 3) + 2)) / ((sqrt(x + 3) - 2) * (sqrt(x + 3) + 2)).

Next, we can simplify further by using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).

Applying this formula, we get (sqrt(x - 1) * (sqrt(x + 3) + 2)) / ((x + 3) - 4).

Simplifying the denominator, we have (sqrt(x - 1) * (sqrt(x + 3) + 2)) / (x - 1).

Now, we can cancel out the common factor of sqrt(x - 1) in the numerator and denominator.

This leaves us with (sqrt(x + 3) + 2) / 1.

Finally, we can substitute x = 1 into the simplified expression.

When x approaches 1, the limit of sqrt(x - 1) / (sqrt(x + 3) - 2) is equal to (sqrt(1 + 3) + 2) / 1, which simplifies to (sqrt(4) + 2) / 1.

Therefore, the limit is equal to 4.

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