# How do you use the limit definition to find the derivative of #f(x)=x^3+1#?

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To find the derivative of ( f(x) = x^3 + 1 ) using the limit definition of a 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^3 + 1 ) into the definition:

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

- Expand and simplify the numerator:

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

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

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

- Cancel out ( h ) in the numerator and denominator:

[ f'(x) = 3x^2 + 3xh + h^2 ]

- Take the limit as ( h ) approaches 0:

[ f'(x) = 3x^2 + 0 + 0^2 ]

[ f'(x) = 3x^2 ]

So, the derivative of ( f(x) = x^3 + 1 ) is ( f'(x) = 3x^2 ).

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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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