# How do you find the slope of the tangent line for #f(x)=1/absx# at x=4?

Near

Now use either the definition or the power rule to get

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To find the slope of the tangent line for ( f(x) = \frac{1}{|x|} ) at ( x = 4 ), we need to compute the derivative of the function and then evaluate it at ( x = 4 ).

First, we find the derivative of ( f(x) ) using the quotient rule:

[ f'(x) = \frac{d}{dx} \left( \frac{1}{|x|} \right) = \frac{d}{dx} \left( \frac{1}{x} \right) ]

[ = \frac{-1}{x^2} ]

Now, we evaluate ( f'(x) ) at ( x = 4 ):

[ f'(4) = \frac{-1}{4^2} = \frac{-1}{16} ]

So, the slope of the tangent line for ( f(x) = \frac{1}{|x|} ) at ( x = 4 ) is ( -\frac{1}{16} ).

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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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