How do you find f'(x) using the limit definition given #f(x) = 3/(x2)#?
The limit definition of a derivative states that
From this point on, you want to expand and simplify.
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To find ( f'(x) ) using the limit definition given ( f(x) = \frac{3}{x2} ), follow these steps:

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

Substitute ( f(x) = \frac{3}{x2} ) into the definition: [ f'(x) = \lim_{{h \to 0}} \frac{\frac{3}{x+h2}  \frac{3}{x2}}{h} ]

Simplify the expression: [ f'(x) = \lim_{{h \to 0}} \frac{\frac{3(x2)}{(x+h2)(x2)}  \frac{3(x+h2)}{(x2)(x+h2)}}{h} ]

Combine the fractions and simplify further: [ f'(x) = \lim_{{h \to 0}} \frac{3(x2)  3(x+h2)}{h(x2)(x+h2)} ]

Expand and simplify the numerator: [ f'(x) = \lim_{{h \to 0}} \frac{3x  6  3x  3h + 6}{h(x2)(x+h2)} ] [ f'(x) = \lim_{{h \to 0}} \frac{3h}{h(x2)(x+h2)} ]

Cancel out the ( h ) in the numerator and denominator: [ f'(x) = \lim_{{h \to 0}} \frac{3}{(x2)(x+h2)} ]

Evaluate the limit as ( h ) approaches ( 0 ): [ f'(x) = \frac{3}{(x2)(x2)} ]

Simplify the expression: [ f'(x) = \frac{3}{(x2)^2} ]
Thus, ( f'(x) = \frac{3}{(x2)^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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