Is this graph defined at x = 0?

[PcumP_Ravenclaw]

Hello everyone,
Is the graph $(2^x - 1) / x$ defined at x = 0?
When I plot this graph there is no spike at 0 because 0/0 is undefined? Is the computer unable to show this? I am finding the limit as x approaches 0. From the graph limit exits but does f(0) exist?

Sehr Danke.

 

[Drakkith]

When I plot this graph there is no spike at 0 because 0/0 is undefined? Is the computer unable to show this?

The function f(x)=1/x is also undefined at x=0, but there's a spike in the graph of that function because the limit as x approaches 0 is infinity (or negative infinity if coming from the negative side of zero). There's no spike in (2^x-1)/x because the limit as x approaches zero is not infinity.

 

[HallsofIvy]

I would say that it is not a question of the "graph" being defined but the function itself. $f(x)= (2^x- 1)/x$ is "not defined" because "0/0" does not correspond to a number. There is what is usually called a "removable" discontinuity at x= 0. "Removable" because the limit, as x goes to 0, of f(x) exists- it is ln(2). The function "$g(x)= (2^x- 1)/x$ if x is not 0, g(0)= ln(x)" is continuous for all x and is exactly the same as f for all x except 0.