How can you remove a discontinuity?

Answer 1

Please see below.

A discontinuity at #x=c# is said to be removable if
#lim_(xrarrc)f(x)# exists. Let's call it #L#.
But #L != f(c)# (Either because #f(c)# is some number other than #L# or because #f(c)# has not been defined.
We "remove" the discontinuity by defining a new function, say #g(x)#
by #g(x) = {(f(x),"if",x != c),(L,"if",x = c):}#.
We now have #g(x) = f(x)# for all #x != c# and #g# is continuous at #c#,

Example

#f(x) = (x^2-1)/(x-1)# is discontinuous at #x=1#. (#f(1)# does not exist)
But #lim_(xrarr1)f(x) = lim_(xrarr1)(x^2-1)/(x-1)#
# = lim_(xrarr1) (x+1) = 2#

So we remove the discontinuity by defining:

#g(x) = {((x^2-1)/(x-1),"if",x != 1),(2,"if",x = 1):}#.
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Answer 2

To remove a discontinuity in a function, you can use a process called "removable discontinuity removal" or "remediation." This involves finding the limit of the function as it approaches the point of discontinuity and then redefining the function at that point to make it continuous. This can be done by either algebraically simplifying the function or by introducing a new term that cancels out the discontinuity.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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