# What is the slope of #f(x)=(x+2)/e^(x-x^2) # at #x=-1#?

The slope at

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To find the slope of the function ( f(x) = \frac{x + 2}{e^{x - x^2}} ) at ( x = -1 ), you need to compute the derivative of the function and then evaluate it at ( x = -1 ). The derivative of the function ( f(x) ) with respect to ( x ) is:

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

Now, substitute ( x = -1 ) into ( f'(x) ) to find the slope at ( x = -1 ):

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

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

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

[ f'(-1) = \frac{3e^{-2} - 3e^{-2}}{e^{-4}} ]

[ f'(-1) = \frac{0}{e^{-4}} ]

[ f'(-1) = 0 ]

Therefore, the slope of the function ( f(x) = \frac{x + 2}{e^{x - x^2}} ) at ( x = -1 ) is ( 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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