# Is #f(x)=(3-e^(2x))/x# increasing or decreasing at #x=-1#?

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To determine if ( f(x) = \frac{{3 - e^{2x}}}{x} ) is increasing or decreasing at ( x = -1 ), we can use the first derivative test.

First, find the derivative of ( f(x) ):

[ f'(x) = \frac{{d}}{{dx}} \left( \frac{{3 - e^{2x}}}{x} \right) ]

Using the quotient rule,

[ f'(x) = \frac{{(x)(0) - (3 - e^{2x})(1)}}{{x^2}} = \frac{{e^{2x} - 3}}{{x^2}} ]

Now evaluate ( f'(-1) ):

[ f'(-1) = \frac{{e^{-2} - 3}}{{(-1)^2}} = \frac{{e^{-2} - 3}}{1} = e^{-2} - 3 ]

Since ( e^{-2} ) is less than 1, ( e^{-2} - 3 ) is negative. Thus, ( f'(-1) ) is negative.

Because ( f'(-1) ) is negative, ( f(x) ) is decreasing at ( x = -1 ).

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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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