Q. 1 - Calculus

Question

Explain why it makes intuitive sense that $\underset{x\to c}{lim} x=c$ for any real number c. Then use a delta–epsilon argument to prove it. 

Step-by-Step Solution

Answer

The explanation has been given.

Step 1. Given information

We have to explain why it makes intuitive sense that $\underset{x\to c}{lim} x=c$ for any real number c. Then use a delta–epsilon argument to prove it. 

Step 2. Explanation

Substitute c for x in the limit,

$\underset{x\to c}{lim} x=c$

For the limit statement $\underset{x\to c}{lim} f^{\mathrm{\prime }}\left(x\right)=L$, the delta-epsilon statement is

For all $\in >0$, there exist $\delta >0$ such that whenever $x\in (c-\delta ,c)\cup (c,c+\delta )\text{ guarantees }f\left(x\right)\in (L-\in ,L+\in )$

Fof every x satisfying $0<|x-c|<\delta$, every f(x) satisfies $\left|f\right(x)-L|<ϵ$

$f\left(x\right)=x \text{ and }L=c$

Choose $\delta =\in$

$\begin{array}{c}\left|f\right(x)-L|=|x-c|\\ <\delta \\ =\in \end{array}$