# How do you find f'(x) using the limit definition given # f(x)=3x^(−2)#?

I found

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by definition

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To find ( f'(x) ) using the limit definition, we use the formula:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Given ( f(x) = 3x^{-2} ), substitute into the formula:

[ f'(x) = \lim_{h \to 0} \frac{3(x + h)^{-2} - 3x^{-2}}{h} ]

Simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{3}{h} \left( \frac{1}{(x + h)^2} - \frac{1}{x^2} \right) ]

[ = \lim_{h \to 0} \frac{3}{h} \left( \frac{x^2 - (x + h)^2}{x^2(x + h)^2} \right) ]

[ = \lim_{h \to 0} \frac{3}{h} \left( \frac{x^2 - (x^2 + 2xh + h^2)}{x^2(x + h)^2} \right) ]

[ = \lim_{h \to 0} \frac{3}{h} \left( \frac{-2xh - h^2}{x^2(x + h)^2} \right) ]

[ = \lim_{h \to 0} \frac{3(-2x - h)}{x^2(x + h)^2} ]

Now, we can substitute ( h = 0 ) into the expression:

[ f'(x) = \frac{3(-2x)}{x^2 \cdot x^2} ]

[ = \frac{-6}{x^3} ]

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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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