# How do you find the vertical, horizontal and slant asymptotes of: #y = (3e^x )/(e^(x-2))#?

None. The equation represents a straight line

This represents a straight line parallel to the x-axis.

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To find the vertical asymptotes of the function ( y = \frac{3e^x}{e^{x-2}} ), set the denominator equal to zero and solve for ( x ). In this case, the denominator ( e^{x-2} ) will never be zero, since the exponential function is always positive for any real value of ( x ). Therefore, there are no vertical asymptotes for this function.

To find the horizontal asymptote, compare the degrees of the numerator and the denominator when simplified. In this case, as ( x ) approaches positive or negative infinity, the term ( e^{x-2} ) dominates the expression. Thus, the horizontal asymptote is the ratio of the leading coefficients, which is ( 3/e^2 ).

To find the slant asymptote, divide the numerator by the denominator using long division or synthetic division. The quotient will represent the slant asymptote. In this case, dividing ( 3e^x ) by ( e^{x-2} ) yields ( 3e^2 ). Therefore, the slant asymptote is the line ( y = 3e^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.

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