# How do you find the linearization of #f(x) = 2 ln(x)# at the point a=1?

The linearization is the equation of the tangent line (often written in a particular form).

We get the linearization (and the tangent line) using

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To find the linearization of ( f(x) = 2 \ln(x) ) at the point ( a = 1 ), we follow these steps:

- Find the first derivative of ( f(x) ), denoted as ( f'(x) ).
- Evaluate ( f'(1) ) to find the slope of the tangent line at ( x = 1 ).
- Use the point-slope form of a line, ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) ) is the point ( (1, f(1)) ) and ( m ) is the slope we found earlier, to write the equation of the tangent line.

Let's go through these steps:

- The derivative of ( f(x) = 2 \ln(x) ) is ( f'(x) = \frac{2}{x} ).
- Evaluate ( f'(1) ): ( f'(1) = \frac{2}{1} = 2 ).
- Using the point-slope form with ( (x_1, y_1) = (1, 2 \ln(1)) = (1, 0) ) and ( m = 2 ), the equation of the tangent line is ( y - 0 = 2(x - 1) ).

Simplify the equation to get the linearization of ( f(x) ) at ( a = 1 ): ( y = 2x - 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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