Formal Definition of a Limit at a Point

Key Questions

• How do you use the limit definition to prove a limit exists?

  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#

  F. Javier B. · 2 · Jun 10 2018
• How do you use the epsilon delta definition of limit to prove that #lim_(x->1)(x+2)= 3# ?

  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#

  Wataru · · Sep 24 2014
• What is the formal definition of limit?

  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#

  Wataru · 1 · Sep 17 2014

• How do you use the epsilon delta definition of limit to prove that #lim_(x->5)(x-1)= 4# ?
• How do you use the epsilon delta definition of limit to prove that #lim_(x->1)(x+2)= 3# ?
• What is the formal definition of limit?
• How do you use the limit definition to prove a limit exists?
• What is the definition of limit in calculus?
• How do you find the limit using the epsilon delta definition?
• How do you use the epsilon delta definition to prove a limit exists?
• What is the epsilon delta definition of limit?
• How do you find values of δ that correspond to ε=0.1, ε=0.05, and ε=.01 when finding the limit of #(5x-7)# as x approaches 2?
• How do you prove that the limit of #3x+5=35# as x approaches 10 using the precise definition of a limit?
• What is an example of a function where the limit as x approaches 3 is the same as the function value at x=3?
• How do you use the formal definition of a limit to find #1/(x - 3) = 0# as x approaches infinity?
• How do you use the formal definition of a limit to find #lim 4x -1# as x approaches 2?
• How do you use the formal definition of a limit to prove #lim (x/(x-3)) =1# as x approaches infinity?
• How do you use the formal definition of a limit to prove #lim (x^2-x+1)=1# as x approaches 1?
• How do you use the formal definition of a limit to prove #lim(x^3 + 3x + 5) = 19# as x approaches 2?
• How to determine which of the following regions has an area equal to the given limit without evaluating the limit ?
• Question #ba263
• How do you prove that the limit of #5 - 2x# as x approaches 2 is equal to 1 using the epsilon delta proof?
• How do you prove that the limit of #(2x^2 + 1) = 3 # as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of #(x^2 - x) = 0 # as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of #(x+2)/(x-3) = -1/4# as x approaches -1 using the epsilon delta proof?
• How do you prove that the limit of #(2-x)/(1+x^2) = 3/2# as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of #x^2 = 9# as x approaches 3 using the epsilon delta proof?
• How do you prove that the limit of #sqrtx = 3# as x approaches 9 using the epsilon delta proof?
• How do you prove that the limit of #(x^3 - 3) = 5# as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #x^(-1/2) = 2# as x approaches 1/4 using the epsilon delta proof?
• How do you prove that the limit of #x-2 = 2# as x approaches 3 using the epsilon delta proof?
• How do you prove that the limit of #(x^2)sin(1/x)=0 # as x approaches 0 using the epsilon delta proof?
• How do you prove that the limit of #(3x+2)=8 # as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #sqrt(x+1) = 2 # as x approaches 3 using the epsilon delta proof?
• How do you prove that the limit of #sqrt(x+7)=3 # as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #(x^2) =9 # as x approaches -3 using the epsilon delta proof?
• How do you prove that the limit of # (x²-9) / (x²+5x+6)=0 # as x approaches -3 using the epsilon delta proof?
• How do you prove that the limit of #x^2 - 5x + 6 = 2# as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of #1/(x-3)# does not exist as x approaches 3 using the epsilon delta proof?
• How do you prove that the limit of #(3x − 5) = 1# as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #x^2 = 0# as x approaches 0 using the epsilon delta proof?
• How do you prove that the limit of #(x^2 - 4x + 5) = 1# as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #(x^2 - 1) = 3# as x approaches -2 using the epsilon delta proof?
• How do you prove that the limit of #(4 + x - 3x^3) = 2 # as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of #5x^2 =5# as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of #((x/4)+3) = 9/2# as x approaches 6 using the epsilon delta proof?
• How do you prove that the limit of #((x^2+x-6)/(x-2))=5# as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #((9-4x^2)/(3+2x))=6# as x approaches -1.5 using the epsilon delta proof?
• How do you prove that the limit of #(x^2+3x)=-18# as x approaches -3 using the epsilon delta proof?
• How do you prove that the limit of #5 - 2x= 1# as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #x+2y= 1# as (x,y) approaches (1,0) using the epsilon delta proof?
• How do you prove that the limit of #(1-5x) = 16# as x approaches -3 using the epsilon delta proof?
• How do you prove that the limit of #(x+3)=5# as x approaches 2 using the epsilon delta proof?
• How do you prove that the limit of #x^(3) = -8# as x approaches -2 using the epsilon delta proof?
• How do you prove that the limit of #f(x) =2x-3# as x approaches 5 is 7 using the epsilon delta proof?
• How do you prove that the limit of #f(x) = 2 - (1/x)# as x approaches 1 using the epsilon delta proof?
• How do you prove that the limit of # (x^3*y^2)/(x^2+y^2)# as (x,y) approaches (0,0) using the epsilon delta proof?
• Question #a44d8
• How do you prove that the limit # 3/sqrt(x-5) = ∞# as x approaches #5^+# using the formal definition of a limit?
• How do you prove that the limit # (x^2 - 4x + 5) = 1# as x approaches 2 using the formal definition of a limit?
• How do you prove that the limit #(x^2 - 1) = 3# as x approaches -2 using the formal definition of a limit?
• How do you prove that the limit # (1-9x^2) / (1-3x) = 2# as x approaches 1/3 using the formal definition of a limit?
• How do you prove that the limit # ((x/4)+3) = 9/2# as x approaches 6 using the formal definition of a limit?
• How do you prove that the limit #((x^2+x-6)/(x-2))=5# as x approaches 2 using the formal definition of a limit?
• How do you prove that the #lim_(x to -2)(x^2 -5) = -1# using the formal definition of a limit?
• How do you prove that the limit #(x^2) = 4# as x approaches 2 using the formal definition of a limit?
• How do you prove that the #lim_(xrarr3) (4x-5)=7# using the formal definition of a limit?
• How do you prove that the limit #x^2 cos(1/x) = 0# as x approaches 0 using the formal definition of a limit?
• How do you prove that the limit #(x^2+x-4)=8# as x approaches 3 using the formal definition of a limit?
• How do you prove that the limit #x^(1/2) = 4# as x approaches 16 using the formal definition of a limit?
• How do you prove that the triad of graphs of #y = cosh x , y = sinh x# and ( their bisector in the y-direction ) #y =(1/2) e^x# becomes asymptotic, as #x to oo#?
• The graphs of the triad #y = cosh x, y = sinh x and y = e^x# meet x = 1 at (1, cosh 1), (1, sinh 1) and (1, e). How do you find these points for the FCF-triad #y=cosh(x(1+1/y)), y=sinh(x(1+1/y)) and y=e^(x(1+1/y))#?
• Question #40631
• Question #0a300
• How do you prove limit of #14-5x=4# as #x->2# using the precise definition of a limit?
• How do you prove limit of #root3(x)# as #x->0# using the precise definition of a limit?
• How do you prove limit of #x^2-3x=-2# as #x->2# using the precise definition of a limit?
• How do you prove limit of #2/sqrt(x-4)=oo# as #x->4^+# using the precise definition of a limit?
• How do you use the epsilon delta definition to prove that the limit of #2x-4=6# as #x->1#?
• How do you use the epsilon delta definition to prove that the limit of #x^2-4# as #x->2#?
• How do you use the epsilon delta definition to prove that the limit of #x^2-7x+3=-7# as #x->2#?
• How do you use the epsilon delta definition to prove that the limit of #x^3=27# as #x->3#?
• How do you use the epsilon delta definition to prove that the limit of #x^3+6x^2=32# as #x->2#?
• How do you use the epsilon delta definition to prove that the limit of #sqrt(x+6)=9# as #x->3#?
• How do you use the epsilon delta definition to prove that the limit of #(1+x)/(4+x)=1/2# as #x->2#?
• How do you use the epsilon delta definition to prove that the limit of #x/(6-x)=1# as #x->3#?
• How do you use the epsilon delta definition to prove that the limit of #sqrtabsx=0# as #x->0#?
• How do you find the Limit of #x+3# as #x->2# and then use the epsilon delta definition to prove that the limit is L?
• How do you find the Limit of #2x+5# as #x->-3# and then use the epsilon delta definition to prove that the limit is L?
• How do you find the Limit of #1/2x-1# as #x->-4# and then use the epsilon delta definition to prove that the limit is L?
• How do you find the Limit of #-1# as #x->2# and then use the epsilon delta definition to prove that the limit is L?
• How do you find the Limit of #abs(x-2)# as #x->-2# and then use the epsilon delta definition to prove that the limit is L?
• How do you find the Limit of #x^2+1# as #x->1# and then use the epsilon delta definition to prove that the limit is L?
• Is the statement "if f is undefined at x=c, then the limit of f(x) as x approaches c does not exist" a true or false statement?
• Is the statement "if the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)<0.001" a true or false statement?
• Is the statement "if f(c)=L, then the limit of f(x)=L as x approaches c" a true or false statement?
• Is the statement "if limit of f(x)=L as x approaches c, then f(c)=L" a true or false statement?
• Consider the line f(x)=mx+b, where m does not equal 0, how do you use the epsilon delta definition of a limit to prove that the limit f(x)=mc+b as x approaches c?
• How do you evaluate limit of #(x^a-1)/(x^b-1)# as x approaches 1?
• Question #270be
• In the limit #lim 10x=40# as #x->4#, how do you find #delta>0# such that whenever #0<abs(x-4)<delta#, #abs(10x-40)<0.01#?
• In the limit #lim (x^2-4x)/(2x)=-2# as #x->4#, how do you find #delta>0# such that whenever #0<abs(x)<delta#, #abs((x^2-4x)/(2x)-(-2))<0.01#?
• In the limit #lim sqrtx=0# as #x->0^+#, how do you find #delta>0# such that whenever #0<x<delta#, #sqrtx<0.01#?
• In the limit #lim sqrt(x^2-4)=0# as #x->2^+#, how do you find #delta>0# such that whenever #2<x<2+delta#, #sqrt(x^2-4)<0.01#?
• In the limit #lim sqrt(6-3x)=0# as #x->2^-#, how do you find #delta>0# such that whenever #2-delta<x<2#, #sqrt(6-3x)<0.01#?
• In the limit #lim x^-2=oo# as #x->0#, how do you find #delta>0# such that whenever #0<absx<delta#, #x^-2>10,000#?
• In the limit #lim 16/x^4=oo# as #x->0#, how do you find #delta>0# such that whenever #0<absx<delta#, #16/x^4>10,000#?
• In the limit #lim 1/(10t)=oo# as #t->0^+#, how do you find #delta>0# such that whenever #0<t<delta#, #1/(10t)>100#?
• In the limit #lim 1/(4-t)=-oo# as #t->4^+#, how do you find #delta>0# such that whenever #4<t<t+delta#, #1/(4-t)<-100#?
• In the limit #lim 1/x^3=oo# as #x->0^+#, how do you find #delta>0# such that whenever #0<x<delta#, #1/x^3>1000#?
• In the limit #lim_(x->1^-)1/(1-x^2)=oo#, how do you find #delta>0# such that whenever #1-delta<x<1#, #1/(1-x^2)>100#?
• In the limit #lim 5/sqrt(2-x)=oo# as #x->2^-#, how do you find #delta>0# such that whenever #2-delta<x<2#, #5/sqrt(2-x)>100#?
• In the limit #lim_(t to oo) 1/(1+4t)=0# , how do you find #B>0# such that whenever #t>B#, #1/(1+4t)<0.01#?
• Is the following statement True or False: " f'(x) exists for all real numbers x and f'(x) is never 0, then f(1) can not be equal to f(5). "?
• Find the limit... ?
• Find the limit... ?
• Find the limit... ?
• Question #f7df0
• Question #b285a
• Find the following limits if they exist. #lim_(x->5)(|10-2x|)/(3x^2-13x-10)#? #lim_(x->1)(sqrtx-1)/(root(3)x -1)#? Please help
• Verify the following limit?