Question

Find the indicated limit, if it exists.

Answer 1

Plug x = 1 into the first piece
f(x) = 1-x
f(1) = 1-1
f(1) = 0
So as x approaches 1 from the left, it approaches y = 0. This is the left hand limit (LHL).

But as x approaches 1 from the right, it will approach y = 8 since plugging x = 1 into the third piece leads to x+7 = 1+7 = 8. This is the right hand limit (RHL)

We have this disconnect. See the attached image for a visual of what's going on here. This is known as a jump discontinuity. This jump indicates that the limit does NOT exist at x = 1. The LHL (found above to be 0) is not equal to the RHL (found above to be 8). The LHL and RHL must be the same value for the overall limit to exist.

Answer: Does Not Exist (DNE)