Find the horizontal and vertical asymptotes of the curve.
y = 7e^x/e^x−8
x=?
y=?(smaller y-value)
y=?(larger y-value)
use the concepts I used in my previous answer to your similar question.
To find the horizontal asymptote of the curve represented by the equation y = 7e^x / (e^x - 8), we need to determine the behavior of the curve as x approaches positive or negative infinity.
Horizontal Asymptote:
1. As x approaches positive infinity (x → +∞), e^x also approaches positive infinity. Therefore, the term e^x in the denominator becomes significantly larger compared to the constant term 8. So, we can ignore the 8 in the denominator.
y = 7e^x / (e^x - 8)
≈ 7e^x / e^x (as x → +∞)
= 7
Thus, the horizontal asymptote of the curve is y = 7.
Vertical Asymptote:
To find vertical asymptotes, we look for values of x for which the denominator of the fraction becomes zero.
2. e^x - 8 = 0
e^x = 8
x = ln(8)
Therefore, there is a vertical asymptote at x = ln(8).
Now, let's determine the y-values for smaller and larger x-values.
For smaller y-values:
To find the y-value when the x-value is approaching negative infinity (x → -∞), we can use the horizontal asymptote.
As x approaches negative infinity, y approaches y = 7.
For larger y-values:
To find the y-value when the x-value is approaching positive infinity (x → +∞), we already found that y approaches y = 7.
As x approaches positive infinity, y also approaches y = 7.
Therefore, the smaller and larger y-values on the curve approach y = 7 as x approaches negative or positive infinity respectively.