# How do you find the limit of #(sqrt(x+4) -2) / x# as x approaches 0?

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To find the limit of (sqrt(x+4) -2) / x as x approaches 0, we can use algebraic manipulation and the concept of limits.

First, let's simplify the expression by multiplying both the numerator and denominator by the conjugate of the numerator, which is (sqrt(x+4) + 2). This will help us eliminate the square root in the numerator.

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

Simplifying this expression gives us:

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

Further simplifying, we get:

(x) / (x * (sqrt(x+4) + 2))

Now, we can cancel out the x terms:

1 / (sqrt(x+4) + 2)

As x approaches 0, the expression becomes:

1 / (sqrt(0+4) + 2) = 1 / (2 + 2) = 1 / 4

Therefore, the limit of (sqrt(x+4) -2) / x as x approaches 0 is 1/4.

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