Classifying Topics of Discontinuity (removable vs. non-removable)
Key Questions
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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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#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^2-1}/{x-1}# has a removable discontinuity at#x=1# since#lim_{x to 1}{x^2-1}/{x-1} =lim_{x to 1}{(x+1)(x-1)}/{x-1} =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
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Introduction to Limits
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Determining One Sided Limits
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Determining When a Limit does not Exist
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Determining Limits Algebraically
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Infinite Limits and Vertical Asymptotes
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Limits at Infinity and Horizontal Asymptotes
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Definition of Continuity at a Point
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Classifying Topics of Discontinuity (removable vs. non-removable)
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Determining Limits Graphically
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Formal Definition of a Limit at a Point
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Continuous Functions
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Intemediate Value Theorem
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Limits for The Squeeze Theorem