4. The following functions are defined for all values of x except x = 0. Which function can be defined at x = 0 so that the function is continuous at x = 0?

 

A. sin(1/x)

B. cos(1/x)

C. x/x2

D. (tan x)/x

E. (x2)1/2/x

 

Solution: The answer is D

 

If any of the above functions can be defined at x = 0 so that the function is continuous at x = 0 then the limit of that function as x approaches zero must exist.

Since,

is the only limit that exists, then tan(x)/x is the only function that can be defined at x = 0 so that the function is continuous at x = 0.

The reason the limit as x ->0 does not exist for the others is:

In A and B the functions oscillate, taking values between +1 and -1 infinitely often as x -> 0.

In C, x/x2 = 1/x, so the limit is + infinity as x -> 0+ and -infinity as x -> 0-.

In E (x2)1/2/x = |x|/x, whose limit is -1 from the left and + from the right.