# What is the arclength of #f(x)=(x-2)/(x^2-x-2)# on #x in [1,2]#?

Arclength is given by:

Integrate directly:

Insert the limits of integration:

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To find the arc length of the function ( f(x) = \frac{x - 2}{x^2 - x - 2} ) on the interval ([1,2]), we first need to compute its derivative and then apply the arc length formula. The derivative of ( f(x) ) is calculated as ( f'(x) = \frac{2x^2 - 6x - 1}{(x^2 - x - 2)^2} ). The arc length formula for a function ( f(x) ) on an interval ([a,b]) is given by ( \int_a^b \sqrt{1 + (f'(x))^2} , dx ). Substituting the derivative into this formula and integrating over the interval ([1,2]), we obtain the arc length of ( f(x) ) on this interval.

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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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