# Given #f(x)=sqrtx# when x=25, how do you find the linear approximation for #sqrt25.4#?

Given:

We have:

So:

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To find the linear approximation for ( \sqrt{25.4} ) using the function ( f(x) = \sqrt{x} ) when ( x = 25 ), follow these steps:

- Start with the given function ( f(x) = \sqrt{x} ).
- Find the derivative of the function ( f(x) ), denoted as ( f'(x) ).
- Evaluate ( f(x) ) and ( f'(x) ) at ( x = 25 ) to get the values ( f(25) ) and ( f'(25) ).
- Use the linear approximation formula:

[ f(a + \Delta x) ≈ f(a) + f'(a) \cdot \Delta x ]

where ( a ) is the known value (in this case, 25), ( f(a) ) is the known square root of that value, and ( \Delta x ) is the difference between the value you're estimating and the known value (in this case, ( \Delta x = 25.4 - 25 = 0.4 )).

- Plug in the values ( f(25) ), ( f'(25) ), and ( \Delta x = 0.4 ) into the linear approximation formula to find the approximation for ( \sqrt{25.4} ).

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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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