# Is #f(x)=sinx/x# increasing or decreasing at #x=pi/3#?

Decreasing.

To determine if a function is increasing or decreasing at a point, use the function's derivative:

graph{sinx/x [-3.945, 4.825, -1.568, 2.817]}

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To determine whether ( f(x) = \frac{\sin x}{x} ) is increasing or decreasing at ( x = \frac{\pi}{3} ), we need to analyze the sign of its derivative at that point. Let's find the derivative of ( f(x) ) with respect to ( x ) using the quotient rule:

[ f'(x) = \frac{(x)(\cos x) - (\sin x)(1)}{x^2} ]

Simplify this expression:

[ f'(x) = \frac{x \cdot \cos x - \sin x}{x^2} ]

Now, evaluate ( f'(x) ) at ( x = \frac{\pi}{3} ):

[ f'\left(\frac{\pi}{3}\right) = \frac{\frac{\pi}{3} \cdot \cos\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{3}\right)}{\left(\frac{\pi}{3}\right)^2} ]

[ f'\left(\frac{\pi}{3}\right) = \frac{\frac{\pi}{3} \cdot \frac{1}{2} - \frac{\sqrt{3}}{2}}{\left(\frac{\pi}{3}\right)^2} ]

[ f'\left(\frac{\pi}{3}\right) = \frac{\frac{\pi}{6} - \frac{\sqrt{3}}{2}}{\left(\frac{\pi}{3}\right)^2} ]

[ f'\left(\frac{\pi}{3}\right) = \frac{\pi - 3\sqrt{3}}{2\pi^2} ]

Since ( \pi - 3\sqrt{3} ) is positive and ( 2\pi^2 ) is positive, ( f'\left(\frac{\pi}{3}\right) ) is positive.

Therefore, ( f(x) = \frac{\sin x}{x} ) is increasing at ( x = \frac{\pi}{3} ).

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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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