# How do you evaluate the limit #(sqrt(x+6)-x)/(x-3)# as x approaches #3#?

Multiply the numerator/denominator by the conjugate of the numerator.

exclusion x ≠ 3

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To evaluate the limit (sqrt(x+6)-x)/(x-3) as x approaches 3, we can use algebraic manipulation. First, we substitute x=3 into the expression:

(sqrt(3+6)-3)/(3-3)

This simplifies to:

(sqrt(9)-3)/0

Since the denominator is 0, we cannot directly evaluate the limit using substitution. However, we can simplify the expression further by rationalizing the numerator:

(sqrt(9)-3)/0 = (3-3)/0 = 0/0

This is an indeterminate form, which means we need to apply additional techniques to evaluate the limit. One approach is to use L'Hôpital's Rule, which states that if we have an indeterminate form of the type 0/0 or ∞/∞, we can differentiate the numerator and denominator separately and then evaluate the limit again.

Differentiating the numerator and denominator, we get:

d/dx (sqrt(x+6)-x) / d/dx (x-3)

Applying the derivative, we have:

(1/2)*(x+6)^(-1/2) - 1 / 1

Now, we substitute x=3 into the derivative expression:

(1/2)*(3+6)^(-1/2) - 1 / 1

Simplifying further:

(1/2)*(9)^(-1/2) - 1 / 1

(1/2)*(1/3) - 1 / 1

1/6 - 1 / 1

1/6 - 6/6

-5/6

Therefore, the limit of (sqrt(x+6)-x)/(x-3) as x approaches 3 is -5/6.

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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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