How do you use the limit definition of the derivative to find the derivative of #f(x)=x+6#?
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To find the derivative of ( f(x) = x + 6 ) using the limit definition of the derivative, follow these steps:

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

Substitute the function ( f(x) = x + 6 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(x  h + 6)  (x + 6)}{h} ]

Simplify the expression inside the limit: [ f'(x) = \lim_{h \to 0} \frac{x  h + 6 + x  6}{h} ] [ f'(x) = \lim_{h \to 0} \frac{h}{h} ]

Cancel out the ( h ) terms: [ f'(x) = \lim_{h \to 0} 1 ]

Evaluate the limit: [ f'(x) = 1 ]
So, the derivative of ( f(x) = x + 6 ) with respect to ( x ) is ( 1 ).
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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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