Performance Authentic
Task 1
Removable Discontinuity
When the limit of a function exists but does not equal the function's value at that point, a removable
discontinuity occurs; this could be due to the function not existing at that point.
Removable vs Non-removable Discontinuity
Geometrically, a removable discontinuity is a hole in the graph of f that may be eliminated.
Non-removable discontinuities are any other types of discontinuities that cannot be removed.(Often jump
or infinite discontinuities.)
Example
Let’s Consider the following piecewise function:
Example
Following is the graph of function:
Example
The function is discontinuous at x = -2, x = 0 and x = 1. But discontinuity at x = -2 and x = 0 are non-removable
discontinuities and discontinuity at x = 1 is removable. These discontinuities can be seen from graph given in previous
slide.
At x = -2 limit does not exist because if we approach from LHS we get -4 and from right hand side we get 0. So limit
does not exist that is discontinuous.
At x = 0 limit does not exist because if we approach from LHS we get 2 and from right hand side we get 0. So limit
does not exist that is discontinuous.
At x = 1 limit exists that is 1 but value of function does not exist we get 0/0 form so discontinuous.
At x = 2 both limit exist and function value exist and both are equal so it is continuous at x = 2.
Example
We can remove discontinuity at x = 1 by some algebra i.e. by factoring out numerator of third piece of the
function. So the new function becomes:
Example
The graph after removing discontinuity at x = 1: