Infinite Limits and Vertical Asymptotes
Key Questions
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The vertical asymptote is a place where the function is undefined and the limit of the function does not exist.
This is because as
#1# approaches the asymptote, even small shifts in the#x# -value lead to arbitrarily large fluctuations in the value of the function.
On the graph of a function
#f(x)# , a vertical asymptote occurs at a point#P=(x_0,y_0)# if the limit of the function approaches#oo# or#-oo# as#x->x_0# .For a more rigorous definition, James Stewart's Calculus,
#6^(th)# edition, gives us the following:"Definition: The line x=a is called a vertical asymptote of the curve
#y=f(x)# if at least one of the following statements is true:#lim_(x->a)f(x) = oo#
#lim_(x->a)f(x) = -oo#
#lim_(x->a^+)f(x) = oo#
#lim_(x->a^+)f(x) = -oo#
#lim_(x->a^-)f(x) = oo#
#lim_(x->a^-)f(x) = -oo# "In the above definition, the superscript + denotes the right-hand limit of
#f(x)# as#x->a# , and the superscript denotes the left-hand limit.Regarding other aspects of calculus, in general, one cannot differentiate a function at its vertical asymptote (even if the function may be differentiable over a smaller domain), nor can one integrate at this vertical asymptote, because the function is not continuous there.
As an example, consider the function
#f(x) = 1/x# .As we approach
#x=0# from the left or the right,#f(x)# becomes arbitrarily negative or arbitrarily positive respectively.In this case, two of our statements from the definition are true: specifically, the third and the sixth. Therefore, we say that:
#f(x) = 1/x# has a vertical asymptote at#x=0# .See image below.
Sources:
Stewart, James. Calculus.#6^(th)# ed. Belmont: Thomson Higher Education, 2008. Print. -
The vertical asymptote of
#y=1/(x+3)# will occur when the denominator is equal to 0. In this case, that will occur at -3, so the vertical asymptote occurs at#x=-3# . There is no y-coordinate to be included.For a more thorough explanation behind vertical asymptotes, see here: http://socratic.org/questions/what-is-a-vertical-asymptote-in-calculus? In summary however, vertical asymptotes occur at
#x# -values where the limit of the function, either overall or from the right or the left, approaches#+-oo# . -
Answer:
An infinite limit is what a functions y value approaches as it approaches infinity or negative infinity
Explanation:
An infinite limit is what a functions y value approaches as the x value approaches infinity or negative infinity
For example
#limxtooo e^x=oo#
#limxto-oo e^x=0#
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