Find all vertical and horizontal asymptotes of the graph of the function. (Enter your answers as a comma-separated list.)
f(x) = 5 − 3x/5 + 8x
To find the vertical asymptotes of a function, we need to check for values of x that make the denominator of the function equal to zero.
For the given function, the denominator is 5 + 8x. To find the vertical asymptote, we need to solve the equation: 5 + 8x = 0.
Solving for x, we have:
8x = -5
x = -5/8
Therefore, the vertical asymptote of the function is x = -5/8.
To find the horizontal asymptote, we need to examine the long-term behavior of the function as x approaches positive or negative infinity.
Let's take the limit of the function as x approaches positive infinity:
Limit as x approaches infinity of (5 - 3x) / (5 + 8x)
To evaluate this limit, we can divide both the numerator and denominator by the highest power of x, which is x:
Limit as x approaches infinity of (5/x - 3) / (5/x + 8)
As x approaches infinity, 5/x approaches 0 (since x gets larger and larger, 5 divided by a large number approaches 0).
Therefore, the limit becomes:
Limit as x approaches infinity of (0 - 3) / (0 + 8)
= (-3) / 8 = -3/8
So, the horizontal asymptote of the function is y = -3/8.
In summary, the vertical asymptote is x = -5/8 and the horizontal asymptote is y = -3/8.
come on -- this is Algebra II
How far do you get?