How do you use the limit definition of the derivative to find the derivative of #f(x)=4/(x3)#?
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To find the derivative of ( f(x) = \frac{4}{x  3} ) using the limit definition of the derivative, follow these steps:

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

Substitute ( f(x) = \frac{4}{x  3} ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{\frac{4}{x + h  3}  \frac{4}{x  3}}{h} ]

Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{4(x  3)  4(x + h  3)}{h(x  3)(x + h  3)} ]

Expand and simplify the numerator: [ f'(x) = \lim_{h \to 0} \frac{4x  12  4x  4h + 12}{h(x  3)(x + h  3)} ] [ f'(x) = \lim_{h \to 0} \frac{4h}{h(x  3)(x + h  3)} ]

Cancel out ( h ) in the numerator and denominator: [ f'(x) = \lim_{h \to 0} \frac{4}{(x  3)(x + h  3)} ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = \frac{4}{(x  3)^2} ]
So, the derivative of ( f(x) = \frac{4}{x  3} ) is ( f'(x) = \frac{4}{(x  3)^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.
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