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?