Finding value "c" so discontinuous function f(x) = [1/(e^1/x -1)]+c has limit at x=0

[fresh]

Finding value "c" so discontinuous function f(x) = [1/(e^1/x -1)]+c has limit at x=0

For which value of c will the function f(x) = [1/(e^1/x -1)]+c have a limit at x=0

I started by taking the right/left side limits

lim f(x) =c
x>0+

lim f(x)=-1+c
x>0-

But I am not sure how to find the value of c that gives this discontinuous function a limit at x=0

 

[ksdhart2]

Okay, so I'm guessing you missed some grouping symbols there. What you wrote is the following:

\(\displaystyle \dfrac{1}{\dfrac{e^{1}}{x}-1}\)

which is continuous at x = 0, although discontinuous at x = e. But what I believe you meant was:

\(\displaystyle \dfrac{1}{e^\frac{1}{x}-1}\)

as that is discontinuous at x = 0.

That said, your process seems right to me. But I wonder if there may have been a typo in the exercise or something, as this is impossible. In order for the limit to exist as x approaches 0, the left- and right-side limits would need to be equal. That boils down to trying to solve c = c - 1, which is never true.

 

[fresh]

Okay, so I'm guessing you missed some grouping symbols there. What you wrote is the following:

\(\displaystyle \dfrac{1}{\dfrac{e^{1}}{x}-1}\)

which is continuous at x = 0, although discontinuous at x = e. But what I believe you meant was:

\(\displaystyle \dfrac{1}{e^\frac{1}{x}-1}\)

as that is discontinuous at x = 0.

That said, your process seems right to me. But I wonder if there may have been a typo in the exercise or something, as this is impossible. In order for the limit to exist as x approaches 0, the left- and right-side limits would need to be equal. That boils down to trying to solve c = c - 1, which is never true.

yes didn't quite group correctly, sorry!
the full question had two parts
a)For which value of c will the function have a limit at x = 0?
b)Using the value for c found in part (b), how would you define f at x = 0 so that f becomescontinuous at x = 0?

could the answer be that there is no possible value of c? but if so, why would there be a part b) asking you to use the value you found in part a)??