Final answer:
The horizontal asymptote of the curve is y=6, as the function's value approaches 6 when x approaches infinity. The vertical asymptote is at x=ln(7), where the denominator equals zero and the function is undefined.
Step-by-step explanation:
To find the horizontal and vertical asymptotes of the curve y = 6ex / (ex − 7), we need to analyze the behavior of the function as x approaches infinity and also when the denominator approaches zero.
For the horizontal asymptote, we look at the limits as x approches ±∞ (positive and negative infinity). Since the degrees of ex in the numerator and denominator are the same, the horizontal asymptote is found by dividing the coefficients, yielding y = 6/1 = 6.
The vertical asymptote occurs where the denominator is zero. In this case, set ex − 7 = 0 and solve for x to get ex = 7 or x = ln(7). Therefore, our vertical asymptote is at x = ln(7).