Calculus Continuity and Discontinuity Piecewise function

Question

f(x) = { 2x - 1 : x < 0 ;;;; a : x = 0 ;;;; x+b/2 : x<0}

a. Determine, if any, the values of a and b that cause f to have a limit for all x but

not continuous for one more x value!

b. Determine, if any, the values of a and b that cause f to be continuous throughout x!

Answer 1

First you have a typo in writing the problem, the last condition is x > 0.

Now the function has a limit and is continuous for x < 0 or X > 0. So we need to verify it at x =0.

Lim f(×) as X approaches to 0 from the left is 2(0) - 1 = -1

and from the right is x + b / 2 = 0 + b/2

These two limits must be equal since the function is supposed to have limit for all x's.

So b/2 = -1 or b = -2

Now if the value of the function is different than this limit -1, then it is not continuous so the answer to part a is b = -2 , a not = -1, meaning a can be any number other than -1, which makes the function to have a limit at 0 but not continuous.

If a = -1 then the function will be also continuous.

There forever the answer for part B is

b= -2 , a = -1