How do you find asymptotic discontinuity?

Answer 1
If a function #f(x)# has a vertical asymptote at #a#, then it has a asymptotic (infinite) discontinuity at #a#. In order to find asymptotic discontinuities, you would look for vertical asymptotes. Let us look at the following example.
#f(x)={x+1}/{(x+1)(x-2)}#
In order to have a vertical asymptote, the function has to display "blowing up" or "blowing down" behaviors. In the case of a rational function like #f(x)# here, it display such behaviors when the denominator becomes zero.

By setting the denominator equal to zero,

#(x+1)(x-2)=0 Rightarrow x=-1,2#

Now, we have a couple of candidates to consider. Let us make sure that there is a vertical asymptote there.

Is #x=-1# a vertical asymptote?
#lim_{x to -1}{(x+1)}/{(x+1)(x-2)}#
by cancelling out #(x+1)#'s,
#=lim_{x to -1}1/{x-2}=1/{1-2}=-1 ne pminfty#,
which means that #x=-1# is NOT a vertical asymptote.
Is #x=2# a vertical asymptote?
#lim_{x to 2^+}{x+1}/{(x+1)(x-2)}#
by cancelling out #(x+1)#'s,
#=lim_{x to 2^+}1/{x-2}=1/0^+=+infty#,
which means that #x=2# IS a vertical asymptote.
Hence, #f# has an asymptotic discontinuity at #x=2#.

I hope that this was helpful.

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Answer 2

To find asymptotic discontinuity, you need to analyze the behavior of a function as it approaches a certain value or values.

  1. Determine the function's domain and any potential points of discontinuity.
  2. Check if the function approaches different values from the left and right sides of the potential discontinuity point(s).
  3. If the function approaches different values from the left and right sides, it indicates a potential asymptotic discontinuity.
  4. Further investigate the behavior of the function by evaluating the limit as it approaches the potential discontinuity point(s) from both sides.
  5. If the limit does not exist or approaches infinity, the function has an asymptotic discontinuity at that point.
  6. If the limit exists and approaches a finite value, the function does not have an asymptotic discontinuity at that point.

Remember to consider both one-sided limits and two-sided limits when analyzing asymptotic 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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