9514 1404 393
Answer:
(i) not continuous at x=2
(ii) -√2
Explanation:
(i) The function simplifies to ((x -2)(x+2))/(x -2) = x +2 . . . . x ≠ 2
The left and right limits at x=2 exist and are the same: 4. However, the function is undefined at x=2, so the conditions for continuity are not met.
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(ii) The expression is indeterminate (0/0) at x=π/4, so L'Hopital's rule can be used to find the limit.
lim[x→π/4] = ((d/dx)(cos(x)-sin(x)))/((d/dx)(x-π/4)) = (-sin(π/4)-cos(π/4))/1
lim[x→π/4] = -√2/2-√2/2 = -√2
The limit is -√2.