Using the limit definition, how do you differentiate #f(x) = 7x + 8#?
See the explanation section below.
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To differentiate the function ( f(x) = 7x + 8 ) using the limit definition, you 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) = 7x + 8 ) into the limit definition: [ f'(x) = \lim_{h \to 0} \frac{(7(x + h) + 8)  (7x + 8)}{h} ]

Simplify the expression inside the limit: [ f'(x) = \lim_{h \to 0} \frac{7x + 7h + 8  7x  8}{h} ] [ f'(x) = \lim_{h \to 0} \frac{7h}{h} ]

Cancel out the ( h ) terms: [ f'(x) = \lim_{h \to 0} 7 ]

Evaluate the limit: [ f'(x) = 7 ]
Therefore, the derivative of ( f(x) = 7x + 8 ) with respect to ( x ) is ( f'(x) = 7 ).
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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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