How do you find the linearization at a=1 of #f(x) = 2 ln(x)#?
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To find the linearization of ( f(x) = 2 \ln(x) ) at ( a = 1 ), follow these steps:

Find the first derivative of ( f(x) ): [ f'(x) = \frac{2}{x} ]

Evaluate the derivative at ( x = a = 1 ): [ f'(1) = \frac{2}{1} = 2 ]

Find the value of ( f(a) ): [ f(1) = 2 \ln(1) = 2 \cdot 0 = 0 ]

Use the formula for the linearization: [ L(x) = f(a) + f'(a)(x  a) ]

Substitute the values of ( f(a) ), ( f'(a) ), and ( a ) into the formula: [ L(x) = 0 + 2(x  1) ]

Simplify: [ L(x) = 2x  2 ]
So, the linearization of ( f(x) = 2 \ln(x) ) at ( a = 1 ) is ( L(x) = 2x  2 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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