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

Question

Sophie Adams · Accepted Answer

None. The equation represents a straight line #y=3e^2=22.17#, nearly

#y=r^x/(e^xe^(-2))=e^2=22.17#, nearly. This represents a straight line parallel to the x-axis.

Sadie Alexander · Answer

undefined 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 $.