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Essential discontinuity is one of the types of discontinuity in the topic of limits. One thing is clear that you will find discontinuity in this topic but the behaviour of discontinuity also matters. That is why mathematicians made different types of discontinuity. Let's say you have a function 
for range 
. The function starts from 
and goes to negative infinity but if the limit is approached from the right to left, we won't get anything because the range of the function starts from , after , there is nothing, you can call it void. Hence limit beyond 2 doesn't exist. There is a discontinuity point at but what kind of discontinuity? Since the limit beyond doesn't exist, we will call it essential discontinuity or infinite discontinuity. There are two conditions for essential discontinuity, if one of them is true, you can declare the limit has an essential discontinuity. Below are the conditions:
- The left or right side limit is infinite
- The left or right side limit do not exist
At 
there is an essential discontinuity because there is no right side limit.
Examples
You might note one more thing in essential discontinuity. Usually, there is a point of discontinuity, but there is an asymptote in the case of essential discontinuity. For example, 
. Let's apply limits to check the discontinuity asymptote.
At there is an essential discontinuity because the right-side limit is infinite. Draw a line on , it becomes an asymptote. Therefore, many mathematicians agree that essential discontinuity has a discontinuous asymptote.
At there is an essential discontinuity because there is no left side limit.
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Correct equals infinity equals 16 but not true. It’s it’s six it’s infinity.
∞ = -1/12
By Sriniwas aramanujan
And this, my friends, is why humans will be conquered by AI… so many logic holes it’s… well, infinite. lol
Is 0^infinity ( zero to the power infinity) indeterminate form? How?
Didn’t Cantor proved that there are a group of infinities (the Aleph zero & Aleph one sets, for example)? And these are grouped around the concept of infinite to the power infinity, if I remember correctly .
Exercise 3
I think it is discontinuous at 0
exercise 2 : the function is not defined for x= 0.
exercise 1 q 5 The function has a jump discontinuity at x = 1, should be x=0.