When x = 2e, \lim_(h \to 0) (ln(x + h) - ln(x))/(h) is ? A. (1)/(2e) B. 1 C. ln(2e) D. nonexistant

Question

When x = 2e,
`\lim_(h \to 0) (ln(x + h) - ln(x))/(h)` is ?

A.
`(1)/(2e)`
B. 1
C. ln(2e)
D. nonexistant

Answer 1

`\lim_(h \to 0 ) ( ln(x + h) - ln(x) )/(h)`

`= (d( ln(x)) )/(dx)`

`= (1)/(x)`

when x = 2e

`= (1)/(2e)`

So , correct option is (A) 1/2e .

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Answer 2

Supposing you know about the derivative, notice that

`\displaystyle\lim_(h\to0)\frac{\ln(x+h)-\ln x}h=(\mathrm d(\ln x))/(\mathrm dx)=\frac1x`

so that when
`x=2e`, the limit is equal to
`\frac1{2e}` and the answer is A.