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Prove that the limit of a constant number is that number. Show that $\begin{equation} \lim_{x\to a}\,K=K \end{equation}$ where $K$ is a constant.

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Lesson Parent:

Problem:

Prove that the limit of a constant number is that number. In other words, prove that

$\begin{equation} \lim_{x\to a}\,K=K \end{equation}$
where

$K$ is a constant.

Answer:

It is true that

$\begin{equation} \lim_{x\to a}\,K=K. \end{equation}$

Solution:

Here are the steps for this problem:

Step	Description
1	Because <Sign in to see all the formulas> all the time, we will choose any value for <Sign in to see all the formulas>.
2	Say <Sign in to see all the formulas>.
3	<Sign in to see all the formulas> is whatever number we want. Let's say <Sign in to see all the formulas> where $D$ is any positive number.
4	Now assume that we are at some value for $x$ such that <Sign in to see all the formulas>.
5	Now plug the function <Sign in to see all the formulas> into <Sign in to see all the formulas> so that we have <Sign in to see all the formulas>.
6	Normally we would do some algebra in this step, but because <Sign in to see all the formulas> and is already less than any value of <Sign in to see all the formulas> then we are done.
7	Since $C$ is arbitrary we have shown that we can choose any <Sign in to see all the formulas> to have <Sign in to see all the formulas>.

Therefore,

Furthermore, the above is true for any value of

$a$ !