# What is the arc length of #f(x)= lnx # on #x in [1,3] #?

To do this we need to apply the formula for the length of the curve mentioned in: How do you find the length of a curve using integration?

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To find the arc length of ( f(x) = \ln(x) ) on the interval ( x ) in ([1,3]), you use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ]

Where ( f'(x) ) is the derivative of ( f(x) ).

For ( f(x) = \ln(x) ), the derivative is ( f'(x) = \frac{1}{x} ).

So, the integral becomes:

[ L = \int_{1}^{3} \sqrt{1 + \left(\frac{1}{x}\right)^2} , dx ]

[ L = \int_{1}^{3} \sqrt{1 + \frac{1}{x^2}} , dx ]

This integral can be evaluated numerically, resulting in the arc length.

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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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