# Is #f(x)=(x-3)/sqrt(x+3) # increasing or decreasing at #x=3 #?

You have to look at the sign of the derivative of the function for

Thus, the function is getting bigger.

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To determine if ( f(x) = \frac{x - 3}{\sqrt{x + 3}} ) is increasing or decreasing at ( x = 3 ), we need to analyze the sign of the derivative of ( f(x) ) at ( x = 3 ). Let's find the derivative of ( f(x) ) first:

( f'(x) = \frac{d}{dx}\left(\frac{x - 3}{\sqrt{x + 3}}\right) )

Using the quotient rule, we get:

( f'(x) = \frac{\sqrt{x + 3} - \frac{1}{2}(x - 3)(x + 3)^{-\frac{1}{2}}}{x + 3} )

Now, let's evaluate ( f'(3) ):

( f'(3) = \frac{\sqrt{3 + 3} - \frac{1}{2}(3 - 3)(3 + 3)^{-\frac{1}{2}}}{3 + 3} )

( f'(3) = \frac{\sqrt{6}}{6} )

Since ( f'(3) > 0 ), the function ( f(x) ) is increasing at ( x = 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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