Is #f(x)=(x-2)/e^x # increasing or decreasing at #x=-2 #?
Increasing. See explanation below.
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To determine if ( f(x) = \frac{x - 2}{e^x} ) is increasing or decreasing at ( x = -2 ), we need to evaluate the derivative of ( f(x) ) at that point.
First, find the derivative of ( f(x) ): [ f'(x) = \frac{(1)(e^x) - (x - 2)(e^x)}{(e^x)^2} ] [ f'(x) = \frac{e^x - xe^x + 2e^x}{e^{2x}} ] [ f'(x) = \frac{(3 - x)e^x}{e^{2x}} ]
Now, evaluate ( f'(-2) ): [ f'(-2) = \frac{(3 - (-2))e^{-2}}{e^{-4}} ] [ f'(-2) = \frac{5e^{-2}}{e^{-4}} ] [ f'(-2) = 5e^2 ]
Since ( f'(-2) ) is positive, ( f(x) ) is increasing at ( x = -2 ).
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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.
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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