How do you use the limit definition of the derivative to find the derivative of #f(x)=3x7#?
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To find the derivative of ( f(x) = 3x  7 ) using the limit definition of the derivative, follow these steps:

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

Substitute the function ( f(x) = 3x  7 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(3(x + h)  7)  (3x  7)}{h} ]

Expand and simplify the expression inside the limit: [ f'(x) = \lim_{h \to 0} \frac{3x + 3h  7  3x + 7}{h} ] [ f'(x) = \lim_{h \to 0} \frac{3h}{h} ]

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

Since the limit of a constant is the constant itself, the derivative is: [ f'(x) = 3 ]
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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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