SOLUTION: Use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x = a. HINT [See Example 2.] (If there is no such value, enter NONE

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Question 1118241: Use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x = a. HINT [See Example 2.] (If there is no such value, enter NONE.)
f(x) = 2-2e^x/x; a=0

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


A couple of preliminary comments....

(1) Do you really think it is any help to us to include "HINT [See Example 2.]" in your message to us?!

(2) The problem as you show it is of little interest, because it is impossible to define f(a) to make f continuous at x=0:

f%280%29+=+2-2e%5E0%2F0+=+2-1%2F0 which is undefined.

If you are working on a problem like this, then your knowledge of mathematics should be sufficient that you would know appropriate use of parentheses is critical.

The function you are asking about is, in fact, f%28x%29+=+%282-2e%5Ex%29%2Fx; not f%28x%29+=+2-2e%5Ex%2Fx.

At x=0, the function value is the indeterminate form 0/0.

So use l'Hospital's rule.

limit as x approaches 0 of %282-2e%5Ex%29%2Fx

is equal to

limit as x approaches 0 of -2e%5Ex%2F1 = -2e^0 = -2

A graphing calculator will confirm this answer.

graph%28400%2C400%2C-5%2C5%2C-10%2C2%2C%282-2e%5Ex%29%2Fx%29