How do you use the limit definition to find the derivative of #f(x)=x/(x+2)#?
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To use the limit definition to find the derivative of ( f(x) = \frac{x}{x+2} ), you can follow these steps:

Begin with the 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+2} ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{\frac{x+h}{x+h+2}  \frac{x}{x+2}}{h} ]

Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{(x+h)(x+2)  x(x+h+2)}{h(x+2)(x+h)} ]

Expand and simplify the numerator: [ f'(x) = \lim_{h \to 0} \frac{x^2 + 2x + hx + 2h  x^2  hx  2x}{h(x+2)(x+h)} ] [ f'(x) = \lim_{h \to 0} \frac{2x + 2h  2x}{h(x+2)(x+h)} ] [ f'(x) = \lim_{h \to 0} \frac{2h}{h(x+2)(x+h)} ]

Cancel out the common factor (2h): [ f'(x) = \lim_{h \to 0} \frac{1}{(x+2)(x+h)} ]

Evaluate the limit as (h) approaches 0: [ f'(x) = \frac{1}{(x+2)^2} ]
So, the derivative of (f(x) = \frac{x}{x+2}) is (f'(x) = \frac{1}{(x+2)^2}).
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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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