# How do you find the limit of #(sqrt(x+3) - sqrt(3)) / x# as x approaches 0?

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To find the limit of (sqrt(x+3) - sqrt(3)) / x as x approaches 0, we can use algebraic manipulation and the concept of limits.

First, we rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator, which is (sqrt(x+3) + sqrt(3)). This gives us:

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

Simplifying this expression, we get:

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

Further simplifying, we have:

x / (x * (sqrt(x+3) + sqrt(3)))

Canceling out the x terms, we are left with:

1 / (sqrt(x+3) + sqrt(3))

Now, as x approaches 0, the expression inside the square root, (x+3), approaches 3. Therefore, the limit of the denominator as x approaches 0 is 2 * sqrt(3).

Substituting this value back into our expression, we get:

1 / (2 * sqrt(3) + sqrt(3))

Simplifying, we have:

1 / (3 * sqrt(3))

Thus, the limit of (sqrt(x+3) - sqrt(3)) / x as x approaches 0 is 1 / (3 * sqrt(3)).

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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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