# How do I us the Limit definition of derivative on #f(x)=ln(x)#?

Let us look at some details.

By Limit Definition,

by rewriting a bit further,

by putting the limit inside the natural log,

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To use the limit definition of the derivative on ( f(x) = \ln(x) ), 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) = \ln(x) ) into the definition.
- Simplify the expression by evaluating ( f(x + h) ) and ( f(x) ).
- Apply properties of logarithms to simplify the expression further.
- Take the limit as ( h ) approaches ( 0 ).
- Evaluate the limit to find the derivative ( f'(x) ).

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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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