Classifying Topics of Discontinuity (removable vs. nonremovable)
Key Questions

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)(x2)}#
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)(x2)=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)(x2)}# by cancelling out
#(x+1)# 's,#=lim_{x to 1}1/{x2}=1/{12}=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)(x2)}# by cancelling out
#(x+1)# 's,#=lim_{x to 2^+}1/{x2}=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.

#f(x)# has a removable discontinuity at#x=a# when#lim_{x to a}f(x)# EXISTS; however,#lim_{x to a}f(a) ne f(a)# . A removable discontinuity looks like a single point hole in the graph, so it is "removable" by redefining#f(a)# equal to the limit value to fill in the hole.

Recall that a function
#f(x)# is continuous at#a# if#lim_{x to a}f(x)=f(a)# ,which can be divided into three conditions:
C1:
#lim_{x to a }f(x)# exists.
C2:#f(a)# is defined.
C3: C1 = C2A removable discontinuity occurs when C1 is satisfied, but at least one of C2 or C3 is violated. For example,
#f(x)={x^21}/{x1}# has a removable discontinuity at#x=1# since#lim_{x to 1}{x^21}/{x1} =lim_{x to 1}{(x+1)(x1)}/{x1} =lim_{x to 1}(x+1)=2# ,but
#f(1)# is undefined. 
#lim_(x>a^)f(x), lim_(x>a^+)f(x)# are finite and#lim_(x>a^)f(x)!=lim_(x>a^+)f(x)# . So it occurs when the left and right limit at#a# do not match, then we say#f(x)# has a jump discontinuity at#a# .This should not be confused with a point discontinuity where:
#lim_(x>a^)f(x)=lim_(x>a^+)f(x)# which means
#lim_(x>a)f(x)# existsand:
#lim_(x>a)f(x)!=f(a)# It could be the case that
#f(a)# is finite or simply DNE.
Questions
Limits

Introduction to Limits

Determining One Sided Limits

Determining When a Limit does not Exist

Determining Limits Algebraically

Infinite Limits and Vertical Asymptotes

Limits at Infinity and Horizontal Asymptotes

Definition of Continuity at a Point

Classifying Topics of Discontinuity (removable vs. nonremovable)

Determining Limits Graphically

Formal Definition of a Limit at a Point

Continuous Functions

Intemediate Value Theorem

Limits for The Squeeze Theorem