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

Approximately

Taking the derivative, we get

So now applying the formula, we get:

Hopefully this helps!

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To find the arc length of ( f(x) = \frac{\sqrt{x^2 - 1}}{x} ) on the interval ([-2, -1]), we use the formula for arc length:

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

where (a = -2) and (b = -1). First, we find ( \frac{dy}{dx} ) by differentiating ( f(x) ):

[ f'(x) = \frac{d}{dx} \left( \frac{\sqrt{x^2 - 1}}{x} \right) ] [ = \frac{d}{dx} \left( \frac{(x^2 - 1)^{1/2}}{x} \right) ] [ = \frac{1}{x} \cdot \frac{1}{2} (x^2 - 1)^{-1/2} \cdot 2x - (x^2 - 1)^{1/2} \cdot \frac{1}{x^2} ] [ = \frac{1}{x} \cdot \frac{x}{\sqrt{x^2 - 1}} - \frac{\sqrt{x^2 - 1}}{x^2} ] [ = \frac{1}{\sqrt{x^2 - 1}} - \frac{\sqrt{x^2 - 1}}{x^2} ]

Now, we can plug this into the arc length formula and integrate:

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

This integral can be challenging to solve analytically, but you can approximate it using numerical methods.

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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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