Definition of a Limit - Maple Help

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Definition of Limit

Main Concept

The precise definition of a limit states that:

Let   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   at $c$ to be $L$, or write

if the following statement is true:

For any e > 0  there is a d > 0 such that whenever

then also

.

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 .)

Example 1



 Prove:    Note: .    Remember you are trying to prove that: For all , there exists a such that: if then . Step1: Determine what to choose for $\mathrm{δ}.$ Substitute all values into .     The relation has been simplified to the form , if you choose $\mathrm{δ}=\frac{\mathrm{ε}}{5}$.   Step 2: Assume  , and use that relation to prove that . $\left|x-c\right|<\mathrm{δ}$ $\left|x-2\right|<\frac{\mathrm{ε}}{5}$ Substitute values for $c$ and $\mathrm{δ}$.  $5\left|x-2\right|<\mathrm{ε}$          



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 $c$: c  =  3. Ask for an $\mathrm{ε}$: = 4. Try to choose $\mathrm{δ}$ small enough so that  implies .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!   $\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{δ}$ =  5. If it's not possible to choose such a $\mathrm{δ}$, the function $f\left(x\right)$ does not have a limit at the point $c$ !





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