Formal Definition of a Limit at a Point
Key Questions
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Answer:
See below
Explanation:
The definition of limit of a sequence is:
Given
#{a_n}# a sequence of real numbers, we say that#{a_n}# has limit#l# if and only if#AA epsilon>0, exists n_0 in NN // AAn>n_0 rArr abs(a_n-l))< epsilon# -
Before writing a proof, I would do some scratch work in order to find the expression for
#delta# in terms of#epsilon# .According to the epsilon delta definition, we want to say:
For all
#epsilon > 0# , there exists#delta > 0# such that
#0<|x-1|< delta Rightarrow |(x+2)-3| < epsilon# .Start with the conclusion.
#|(x+2)-3| < epsilon Leftrightarrow |x-1| < epsilon# So, it seems that we can set
#delta =epsilon# .(Note: The above observation is just for finding the expression for
#delta# , so you do not have to include it as a part of the proof.)Here is the actual proof:
Proof
For all
#epsilon > 0# , there exists#delta=epsilon > 0# such that
#0<|x-1| < delta Rightarrow |x-1|< epsilon Rightarrow |(x+2)-3| < epsilon# -
Precise Definitions
Finite Limit
#lim_{x to a}f(x)=L# if
for all#epsilon>0# , there exists#delta>0# such that
#0<|x-a|< delta Rightarrow |f(x)-L| < epsilon# Infinite Limits
#lim_{x to a}f(x)=+infty# if
for all#M>0# , there exists#delta>0# such that
#0<|x-a|< delta Rightarrow f(x)>M# #lim_{x to a}f(x)=-infty# if
for all#N<0# , there exists#delta>0# such that
#0<|x-a|< delta Rightarrow f(x) < N#
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