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# Undefined limits by direct substitution

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.D (LO)
,
LIM‑1.D.1 (EK)

## Video transcript

let's see if we can figure out the limit of X over natural log of X as X approaches 1 and like always pause this video and see if you can figure it out on your own well we know from our limit properties this is going to be the same thing as the limit as X approaches 1 of x over over the limit the limit as X approaches 1 of the natural log of X now this top limit the one I have in magenta this is pretty straightforward this if we had the graph of y equals x that would be continuous everywhere it's defined for all real numbers and is continuous at all real numbers and so it's continuous the limit as X approaches 1 of X is just going to be this evaluated at x equals 1 so this is just going to be 1 we just put a 1 in for this X so the numerator here would just evaluate to a 1 and then the denominator natural log of X is not defined for all X's and therefore it isn't continuous everywhere but it is continuous at x equals 1 and since it is continuous at x equals 1 then the limit here is just going to be the natural log evaluated at x equals 1 so this is just going to be the natural log the natural log of 1 which of course is 0 e to the 0 power is 1 so this is all going to be equal to this is going to be equal to if we just evaluate it one over one over zero and now we face a bit of a conundrum 1 over 0 is not defined if it was 0 over 0 we wouldn't necessarily be done yet that's indeterminate form as we will learn in the future there are tools we can apply when we're trying to find limits and we evaluate it like this and we get 0 over 0 but 1 over 0 this is undefined which tells us that this limit does not exist so does not exist and we are done
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