Definition of Limit
The precise definition of a limit states that:
Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number.
Define the limit of f at c to be L, or write
limx→cfx = L
if the following statement is true:
For any e > 0 there is a d > 0 such that whenever
0 < x−c < δ
f(x)−L < ε .
Suppose you want to prove that a certain function has a limit. What exactly needs to be determined?
An input range in which there is a corresponding output. (A positive d so that x−c < δ ⇒ f(x)−L < ε.)
limx→25 x − 1 = 9
Note: c = 2 , L =9, fx = 5 x−1 .
Remember you are trying to prove that:
For all ε > 0, there exists a δ >0 such that:
if 0 < x − 2 < δ then 5 x−1 −9 < ε.
Step1: Determine what to choose for δ.
f(x)−L < ε
5 x−1 −9 < ε
Substitute all values into f(x)−L < ε.
5 x−10 < ε
5 x−2 < ε
x−2 < ε5
The relation has been simplified to the form x−c < δ, if you choose δ=ε5.
Step 2: Assume x−c < δ, and use that relation to prove that f(x)−L < ε.
Substitute values for c and δ.
5 x−1− 9 < ε
Follow the instructions, using different functions f, values of c, e and d to observe graphically why the proof works.
1. Choose a function:LinearQuadraticCubicInverseStep
2. Choose a value for c:
3. Ask for an ε:
4. Try to choose δ small enough so that x−c < δ implies fx−L < ε.T. If the blue strip is a river, and the purple strip is a bridge, then the function (green) must only cross the river where the bridge is!
5. If it's not possible to choose such a δ, the function fx does not have a limit at the point c !
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