How do you find the asymptotes for #f(x)=2 * 3^(x-4)#?
graph{y=2*3^(x-4) [-10, 10, -5, 5]}
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To find the asymptotes of the function ( f(x) = 2 \cdot 3^{x-4} ), we look for horizontal and vertical asymptotes.
Horizontal asymptote: There is no horizontal asymptote for this function because the exponential function ( 3^{x-4} ) grows without bound as ( x ) approaches positive or negative infinity.
Vertical asymptote: There are no vertical asymptotes for this function because exponential functions do not have vertical asymptotes. They can only have horizontal asymptotes.
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To find the asymptotes of the function ( f(x) = 2 \cdot 3^{x-4} ), we need to examine both horizontal and vertical asymptotes.
Vertical asymptotes occur where the function approaches positive or negative infinity as ( x ) approaches a certain value. In this case, since ( 3^{x-4} ) is never zero, there are no vertical asymptotes.
Horizontal asymptotes occur as ( x ) approaches positive or negative infinity, and they indicate the long-term behavior of the function. To find horizontal asymptotes, we consider the behavior of the function as ( x ) approaches positive or negative infinity. Since the base of the exponential function is ( 3 ), which is greater than 1, the function grows without bound as ( x ) approaches positive infinity. Therefore, there is a horizontal asymptote at ( y = \infty ). Similarly, as ( x ) approaches negative infinity, the function approaches zero. Therefore, there is a horizontal asymptote at ( y = 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.
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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