# Is #f(x)=1/e^x# increasing or decreasing at #x=0#?

Decreasing

We use the sign of the first derivative to determine whether this is increasing or decreasing at a given point.

A graph of the original function can be examined here:

chart{e^-x [-10, 15.31, -4.05, 8.6]}

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To determine if the function ( f(x) = \frac{1}{e^x} ) is increasing or decreasing at ( x = 0 ), we need to examine the sign of its derivative at that point.

First, let's find the derivative of ( f(x) = \frac{1}{e^x} ) using the chain rule:

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

Now, evaluate ( f'(0) ):

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

Since ( f'(0) = -1 ), which is negative, the function ( f(x) = \frac{1}{e^x} ) is decreasing at ( x = 0 ).

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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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