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

Answer 1
By Limit Definition of the Derivative, #f'(x)=1/x#.

Let us look at some details.

By Limit Definition,

#f'(x)=lim_{h to 0}{ln(x+h)-lnx}/h#
by the log property #lna-lnb=ln(a/b)#,
#=lim_{h to 0}ln({x+h}/x)/h#

by rewriting a bit further,

#=lim_{h to 0}1/hln(1+h/x)#
by the log property #rlnx=lnx^r#,
#=lim_{h to 0}ln(1+h/x)^{1/h}#
by the substitution #t=h/x# (#Leftrightarrow 1/h=1/{tx}#),
#=lim_{t to 0}ln(1+t)^{1/{t}cdot1/x}#
by the log property #lnx^r=rlnx#,
#=lim_{t to 0}1/xln(1+t)^{1/t}#
by pulling #1/x# out of the limit,
#=1/xlim_{t to 0}ln(1+t)^{1/t}#

by putting the limit inside the natural log,

#1/xln[lim_{t to 0}(1+t)^{1/t}]#
by the definition #e=lim_{t to 0}(1+t)^{1/t}#,
#=1/xlne=1/x cdot 1=1/x#
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Answer 2

To use the limit definition of the derivative on ( f(x) = \ln(x) ), follow these steps:

  1. Start with the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
  2. Substitute the function ( f(x) = \ln(x) ) into the definition.
  3. Simplify the expression by evaluating ( f(x + h) ) and ( f(x) ).
  4. Apply properties of logarithms to simplify the expression further.
  5. Take the limit as ( h ) approaches ( 0 ).
  6. Evaluate the limit to find the derivative ( f'(x) ).
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Answer from HIX Tutor

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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