# How do you find the linear approximation #f(x)=2/x#, #x_0=1#?

An equation of a tangent line is a linear approximation.

The linear approximation is

which you may prefer to write as

Here is a graph showing both. You can zoom in and out and drag the graph around the window. When you navigate away from this answer and return, the graph will start n the same position it has now.

graph{(y-2/x)(y+2x-4)=0 [-0.58, 2.458, 1.252, 2.771]}

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To find the linear approximation of the function ( f(x) = \frac{2}{x} ) at ( x_0 = 1 ), follow these steps:

- Calculate the derivative of the function ( f(x) ).
- Evaluate the derivative at the point ( x = x_0 ).
- Use the derivative evaluated at ( x_0 ) to find the equation of the tangent line to the curve at ( x = x_0 ).
- The equation of the tangent line represents the linear approximation of the function near the point ( x_0 ).

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To find the linear approximation of ( f(x) = \frac{2}{x} ) at ( x_0 = 1 ), you first find the derivative of ( f(x) ) with respect to ( x ), denoted as ( f'(x) ). Then, you evaluate ( f(x_0) ) and ( f'(x_0) ). Finally, you use the formula for linear approximation:

[ L(x) = f(x_0) + f'(x_0)(x - x_0) ]

Substitute the values of ( f(x_0) ) and ( f'(x_0) ) into the formula, and simplify to find the linear approximation function ( L(x) ).

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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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