# Using the limit definition, how do you find the derivative of #f(x) = x/(x+4)#?

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To find the derivative of ( f(x) = \frac{x}{x+4} ) using the limit definition of a derivative, we 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) = \frac{x}{x+4} ) into the definition:

[ f'(x) = \lim_{h \to 0} \frac{\frac{x+h}{x+h+4} - \frac{x}{x+4}}{h} ]

- Simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{(x+h)(x+4) - x(x+h+4)}{h(x+4)(x+h+4)} ]

- Expand the numerator:

[ f'(x) = \lim_{h \to 0} \frac{x^2 + 4x + hx + 4h - x^2 - hx - 4x}{h(x+4)(x+h+4)} ]

- Cancel out like terms:

[ f'(x) = \lim_{h \to 0} \frac{4h}{h(x+4)(x+h+4)} ]

- Simplify further:

[ f'(x) = \lim_{h \to 0} \frac{4}{(x+4)(x+h+4)} ]

- Evaluate the limit:

[ f'(x) = \frac{4}{(x+4)(x+4)} ]

[ f'(x) = \frac{4}{(x+4)^2} ]

So, the derivative of ( f(x) = \frac{x}{x+4} ) with respect to ( x ) is ( f'(x) = \frac{4}{(x+4)^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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