Question 179643
Explain a process for finding a limit
.
Write a concise description of the meaning of lim f(x)=10 as x approaches to 3
<pre><font size = 4 color = "indigo"><b>
{{{matrix(2,4,

lim, f(x), "=", 10,
"x->3","","","")}}}

means, colloqually, for any required degree of tolerance
{{{epsilon}}}, (a positive number usually thought of as very small)
there exists another positive number {{{delta}}} (also usually thought 
of as very small) such that whenever x is less than {{{delta}}} units 
away from {{{3}}}, {{{f(x)}}} will always be less than {{{epsilon}}} units away from {{{10}}}.

To write it concisely,

For any {{{matrix(1,3,epsilon, ">", 0)}}}, there exists a 
{{{matrix(1,3,delta, ">", 0)}}}  such that {{{matrix(1,3,abs(f(x)-10), "<", epsilon)}}} whenever {{{matrix(1,5,0, "<", abs(x-3), "<", delta)}}}.
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Examine the continuity of g(x) = x+3 and the point x=2.
<pre><font size = 4 color = "indigo"><b>
{{{g(x)}}} will be continuous at {{{x=2}}} if {{{g(2)=2+3=5}}} and

{{{matrix(2,4,

lim, g(x), "=", 5,
"x->2","","","")}}}       

Suppose we have any required tolerance, {{{epsilon>0}}}. Then
we need to find a {{{delta>0}}} such that if {{{0<abs(x-2)<delta}}},
then {{{abs(g(x)-5)<epsilon}}}.

{{{abs(g(x)-5)<epsilon}}} if and only if
{{{abs((x+3)-5)<epsilon}}} if and only if
{{{abs(x+3-5)<epsilon}}} if and only if
{{{abs(x-2)<epsilon}}}

So we simply take any {{{d}}} such that 
{{{delta<epsilon}}} and then turn the above 
backwards and we have:

{{{abs(x-2)<delta}}} => {{{abs(x-2)<epsilon}}} =>
{{{abs(x+3-5)<epsilon}}} => {{{abs(g(x)-5)<epsilon}}}

Edwin</pre>