What is the arc length of #f(x)=1/x1/(5x) # in the interval #[1,5]#?
Infinite or arguably undefined, since
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To find the arc length of the function (f(x) = \frac{1}{x}  \frac{1}{5x}) in the interval ([1,5]), you can use the arc length formula:
[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]
 First, find (\frac{dy}{dx}) by taking the derivative of (f(x)).
 Then, plug the derivative into the arc length formula.
 Integrate the resulting expression over the interval ([1,5]).
Here's how you can do it:

Compute the derivative (f'(x)): [ f'(x) = \frac{1}{{x^2}} + \frac{1}{{(5x)^2}} ]

Compute (\left(\frac{dy}{dx}\right)^2): [ \left(\frac{dy}{dx}\right)^2 = \left(\frac{1}{{x^2}} + \frac{1}{{(5x)^2}}\right)^2 ]

Add 1 to (\left(\frac{dy}{dx}\right)^2): [ 1 + \left(\frac{dy}{dx}\right)^2 = 1 + \left(\left(\frac{1}{{x^2}} + \frac{1}{{(5x)^2}}\right)^2\right) ]

Integrate (\sqrt{1 + \left(\frac{dy}{dx}\right)^2}) from 1 to 5: [ L = \int_1^5 \sqrt{1 + \left(\frac{1}{{x^2}} + \frac{1}{{(5x)^2}}\right)^2} , dx ]

Evaluate the integral numerically to find the arc length.
The resulting value of (L) will give you the arc length of the function (f(x)) in the interval ([1,5]).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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