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How to find instantaneous rate of change for #f(x) = ln(x) # when x=0?

Answer 1

#f'(x) = 1/x -> f'(0) = 1/0 = 0#

The instantaneous rate of change is #0#.

Since the derivative is also the definition of the instantaneous rate of change, all we need to do is calculate the derivative using a few fundamental principles, like the ones listed below:

#d/dx[ln u] = (u')/(u)#

Letting #u = x -> du = dx#, and so

#f'(x) = 1/x -> f'(0) = 1/0 = 0#

The instantaneous rate of change is, in this case, #0#.

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Answer 2

Evelyn Agard

To find the instantaneous rate of change for ( f(x) = \ln(x) ) when ( x = 0 ), you need to calculate the derivative of ( f(x) ) at ( x = 0 ). The derivative of ( \ln(x) ) is ( \frac{1}{x} ). Evaluating this derivative at ( x = 0 ), we get:

[ \lim_{x \to 0} \frac{1}{x} ]

This limit is undefined because the function ( \frac{1}{x} ) approaches positive infinity as ( x ) approaches 0 from the right side, and negative infinity as ( x ) approaches 0 from the left side. Therefore, the instantaneous rate of change of ( f(x) = \ln(x) ) at ( x = 0 ) is undefined.

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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.