How do you find the derivative of #f(x)=1/(x1)# using the limit process?
#f'(x) =  (1 ) /((x1)^2 #
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To find the derivative of ( f(x) = \frac{1}{x  1} ) using the limit process, follow these steps:

Begin with the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} )

Substitute the function ( f(x) = \frac{1}{x  1} ) into the definition: ( f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h  1}  \frac{1}{x  1}}{h} )

Combine the fractions into a single fraction: ( f'(x) = \lim_{h \to 0} \frac{(x  1)  (x + h  1)}{h(x  1)(x + h  1)} )

Simplify the numerator: ( f'(x) = \lim_{h \to 0} \frac{x  1  x  h + 1}{h(x  1)(x + h  1)} ) ( f'(x) = \lim_{h \to 0} \frac{h}{h(x  1)(x + h  1)} )

Cancel out the common factor of ( h ): ( f'(x) = \lim_{h \to 0} \frac{1}{(x  1)(x + h  1)} )

Evaluate the limit as ( h ) approaches 0: ( f'(x) = \frac{1}{(x  1)(x  1)} )

Simplify the expression: ( f'(x) = \frac{1}{(x  1)^2} )
Therefore, the derivative of ( f(x) = \frac{1}{x  1} ) using the limit process is ( f'(x) = \frac{1}{(x  1)^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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