How do you use the limit definition to find the derivative of #2sqrtx1/(2sqrtx)#?
Using the lit definition we have:
Now rationalize the numerator of the first term:
So:
For the second term of the sum:
and in the same way as above:
so that:
Finally:
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To find the derivative of the function ( f(x) = \frac{2\sqrt{x}  1}{2\sqrt{x}} ) using the limit definition, follow these steps:

Start with the given function: ( f(x) = \frac{2\sqrt{x}  1}{2\sqrt{x}} ).

Compute ( f(x + h) ), where ( h ) is a small increment.

Substitute ( f(x) ) and ( f(x + h) ) into the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ].

Simplify the expression ( \frac{f(x + h)  f(x)}{h} ).

Take the limit as ( h ) approaches 0 to find the derivative.
Applying these steps will yield the derivative of the given function.
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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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