confusing problem?

Question

1. The function f is defined for all x which is not equal to 0 in interval -1
a) How should f(0) be defined in order that f be continuous for all x in the interval -1
b)With f(0) defined as in part (a), use the definition of the derivative to determine whther f'(0) exists.

Answer 1

As x --> 0 the expression approaches 0/0 which is indeterminate. So use L'Hospital's rule, getting:
(1-2cosx)/cosx = (1-2)/1 = -1
lim x ---> 0

So f(0) must be defined as -1 to make f(x) continuous for all x.

There is a hole at f(0), but we have defined that hole to be -1. I'm not sure what you are asking.

Answer 2

a) you need limits for this part
as x approaches 0, sin 2x and sin x also approach 0.
hence f(x) = 1 as x >>> 0

b) find the derivative and use the logic from step a above to determine the answer. just remember cos x = 1 as x approaches 0