HELP PLEASE!!!

Question

F(x)= (1-e^2x)/(1-e^x). What value must be assigned to f(x) at x=0 to make f(x) continuous at x=0?

Answer 1

For f(x) to be continuous when x = 0, lim_x→0 f(x) = f(0)

 

So, f(0) = lim_x→0[(1-e^2x)/(1-e^x)] (Direct substitution yields 0/0)

 

           = lim_x→0[(1-e^x)(1+e^x)/(1-e^x)]

 

           = lim_x→0(1+e^x) = 2

 

If we define f(0) to be equal to 2, the given function will be continuous at x = 0.  

 

 

Answer 2

The numerator should be factored, then the function can be restated as f(x) = 1 + e^x with the restriction that x is not 0.

Then the limit can be seen to be 2, so original function can be treated as a piecewise function, with f(x) = 2 being the 'piece' that applies when x = 2. (Removable discontinuity).

 

This has the advantage of solving the problem at a point in the class when l'Hopital's rule would not have been available yet.

Answer 3

The limit of the function is indeterminate at x =0.

Use L'Hopital's Rule to evaluate the limit as x→ 0. This will give you the answer that you are looking for.