How do you use the limit definition of the derivative to find the derivative of #f(x)=1/x#?
The aim is to eliminate the h on the denominator otherwise division by zero which is undefined. Manipulate the numerator to obtain h as a factor, to cancel with h on denominator.
combine numerator into a single fraction
We can now 'cancel' h
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To find the derivative of ( f(x) = \frac{1}{x} ) using the limit definition of the derivative:

Start with the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ).

Substitute ( f(x) = \frac{1}{x} ) into the definition.

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

Combine the fractions: ( f'(x) = \lim_{h \to 0} \frac{x  (x + h)}{hx(x + h)} ).

Simplify the numerator: ( f'(x) = \lim_{h \to 0} \frac{h}{hx(x + h)} ).

Cancel out ( h ) from the numerator and denominator: ( f'(x) = \lim_{h \to 0} \frac{1}{x(x + h)} ).

Evaluate the limit as ( h ) approaches 0: ( f'(x) = \frac{1}{x^2} ).
So, the derivative of ( f(x) = \frac{1}{x} ) is ( f'(x) = \frac{1}{x^2} ).
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To use the limit definition of the derivative to find the derivative of ( f(x) = \frac{1}{x} ), follow these steps:
 Start with 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{1}{x} ) into the limit definition:
[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h}  \frac{1}{x}}{h} ]
 Combine the fractions in the numerator:
[ f'(x) = \lim_{h \to 0} \frac{\frac{x  (x + h)}{x(x + h)}}{h} ]
 Simplify the expression in the numerator:
[ f'(x) = \lim_{h \to 0} \frac{\frac{h}{x(x + h)}}{h} ]
 Cancel out the ( h ) terms:
[ f'(x) = \lim_{h \to 0} \frac{1}{x(x + h)} ]
 Now, take the limit as ( h ) approaches 0:
[ f'(x) = \frac{1}{x^2} ]
So, the derivative of ( f(x) = \frac{1}{x} ) with respect to ( x ) is ( f'(x) = \frac{1}{x^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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