How do you use the limit definition of the derivative to find the derivative of #f(x)=7x+1#?
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To use the limit definition of the derivative to find the derivative of ( f(x) = 7x + 1 ), follow these steps:

Write down 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 + 1 ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{(7(x + h) + 1)  (7x + 1)}{h} ]

Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{7x + 7h + 1  7x  1}{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 + 1 ) 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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