How do you find f'(x) using the limit definition given #f (x) = -(2/3x) #?
here that becomes
which is what you would expect :-)
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To find ( f'(x) ) using the limit definition, given ( f(x) = -\frac{2}{3x} ), we use the formula:
[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]
First, we substitute the function into the formula:
[ f'(x) = \lim_{h \to 0} \frac{-\frac{2}{3(x+h)} + \frac{2}{3x}}{h} ]
Then, we simplify the expression:
[ f'(x) = \lim_{h \to 0} \frac{-\frac{2}{3(x+h)} \cdot \frac{3x}{3x} + \frac{2}{3x} \cdot \frac{3(x+h)}{3(x+h)}}{h} ]
[ f'(x) = \lim_{h \to 0} \frac{-2x + 2(x+h)}{3x(x+h)h} ]
[ f'(x) = \lim_{h \to 0} \frac{-2x + 2x + 2h}{3x(x+h)h} ]
[ f'(x) = \lim_{h \to 0} \frac{2h}{3x(x+h)h} ]
[ f'(x) = \lim_{h \to 0} \frac{2}{3x(x+h)} ]
Now, we can plug in ( h = 0 ) to find the derivative:
[ f'(x) = \frac{2}{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.
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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