lim ln(y-1)
y->1+
The answer is negative infinity but how do I show the logic of proving that it's negative infinity? I plugged in "1" to the function and got ln(0) which is obviously impossible? Thanks in advance
so put in ln .01 and ln .00001 etc
or
say
j = ln z
e^j = z
e^j = 0 + a little
e^-oo = 1/e^oo --->0 + a little
So I can just say for example, plug values as y gets closer to 1 and solve for the function and notice that the function is getting smaller and smaller (negative) and therefore negative infinity?
To prove that the limit of ln(y-1) as y approaches 1 from the positive side is negative infinity, we can follow these steps:
1. Start by noting that we are dealing with a one-sided limit, as indicated by the "+". This means we only consider values of y that approach 1 from the positive side.
2. Substitute the limit value, y = 1, into the expression ln(y-1). This gives ln(1-1) = ln(0).
3. ln(0) is undefined because the natural logarithm function is not defined for zero or any negative values. This is the reason why you encountered an issue when plugging in "1" into the function.
4. However, we can still determine the limit by recognizing a property of ln(x). As x approaches 0 from the positive side, ln(x) approaches negative infinity. This is because the natural logarithm function grows more negative as the input gets closer to zero.
5. Applying the property mentioned earlier, we conclude that as y approaches 1 from the positive side, ln(y-1) approaches negative infinity.
Therefore, the limit of ln(y-1) as y approaches 1 from the positive side is negative infinity.