Find all vertical and horizontal asymptotes for the graph of y=lnx/lnx-1
isx=e and y=1
Did you mean:
y=lnx/(lnx-1)
or
y=lnx/ln(x-1)
Clearly you did not mean it the way you typed it, since it
would reduce to y = 0
recall that ln e = 1
so, lnx-1 = 0 when x=1
clearly, when x gets huge, lnx-1 is very close to lnx
so the horizontal asymptote is just y = lnx/lnx = 1
To find the vertical and horizontal asymptotes for the graph of y = ln(x)/ln(x-1), we need to examine the behavior of the function as x approaches certain values.
1. Vertical Asymptotes:
Vertical asymptotes occur when the function approaches infinity or negative infinity at certain x-values. In this case, the vertical asymptote occurs when the denominator of the fraction becomes zero.
So, we need to solve the equation x - 1 = 0 to find the x-value that causes the denominator to become zero:
x - 1 = 0
x = 1
Therefore, the vertical asymptote occurs at x = 1.
2. Horizontal Asymptotes:
Horizontal asymptotes occur when the function approaches a certain value as x approaches positive or negative infinity.
To find the horizontal asymptote, we need to determine the limit of the function as x approaches infinity and negative infinity.
As x approaches infinity:
lim(x→∞) ln(x)/ln(x-1)
The natural logarithm function (ln(x)) approaches positive infinity as x approaches infinity. Similarly, ln(x-1) also approaches positive infinity. Therefore, the ratio ln(x)/ln(x-1) approaches 1 as x approaches infinity.
So, the horizontal asymptote is y = 1 as x approaches positive infinity.
As x approaches negative infinity:
lim(x→-∞) ln(x)/ln(x-1)
In this case, ln(x) and ln(x-1) approach negative infinity as x approaches negative infinity. So, the ratio ln(x)/ln(x-1) approaches 1 as x approaches negative infinity.
Thus, the horizontal asymptote is y = 1 as x approaches negative infinity.
To summarize:
- The vertical asymptote is x = 1.
- The horizontal asymptote is y = 1 as x approaches positive and negative infinity.