# How do you find the tangent line approximation to #f(x)=1/x# near #x=1# ?

To find the derivative of

it is best to rewrite it in power form:

Using the power rule for derivatives:

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To find the tangent line approximation to ( f(x) = \frac{1}{x} ) near ( x = 1 ), we use the formula for the equation of a tangent line:

[ y = f'(a)(x - a) + f(a) ]

where ( f'(a) ) is the derivative of ( f(x) ) evaluated at ( x = a ), and ( f(a) ) is the value of ( f(x) ) at ( x = a ).

First, we find the derivative of ( f(x) = \frac{1}{x} ): [ f'(x) = -\frac{1}{x^2} ]

Then, we evaluate ( f'(x) ) at ( x = 1 ): [ f'(1) = -\frac{1}{1^2} = -1 ]

Next, we find the value of ( f(x) ) at ( x = 1 ): [ f(1) = \frac{1}{1} = 1 ]

Now, we have the slope of the tangent line (( f'(1) = -1 )) and the point (( 1, 1 )) through which the tangent line passes. Using these values, we can write the equation of the tangent line: [ y = -1(x - 1) + 1 ]

[ y = -x + 1 + 1 ]

[ y = -x + 2 ]

So, the tangent line approximation to ( f(x) = \frac{1}{x} ) near ( x = 1 ) is ( y = -x + 2 ).

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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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