Approaching a Limit: Does x Reach 1?

[bacon]

When approaching a limit, say, x approaching 1, does x actually reach 1 or is it just infinitesimally close? In particular I'm interested in where the denominator of a function of interest contains the factor (x-1).

(It's hard to show a nice example without the Latex.)

 

[ImAnEngineer]

Yes, you get as close as you can get.

 

[HallsofIvy]

The definition of "limit" is "lim_{x\rightarrow a}= L if and only if given any \epsilon&gt; 0 there exist \delta&gt; 0 such that if 0&lt; |x- a|&lt; \delta then |f(x)- f(a)|&lt; \epsilon. Notice the "0< |x-a|". What happens at x= a is irrelevant.

For example, if f(x)= x²+ 3 for all x except 1 and f(1)= 100000, then lim_{x\rightarrow 1} f(x)is still 3+ 1= 4: for any value of x &lt;b&gt;close&lt;/b&gt; to 1 but &lt;b&gt;not equal&lt;/b&gt; to 1, f(x) is close to 4.&lt;br /&gt; &lt;br /&gt; Since you mention &amp;quot;In particular I&amp;#039;m interested in where the denominator of a function of interest contains the factor (x-1)&amp;quot;, take f(x)= (x^2- 1)/(x-1). To find lim_{x\rightarrow 1}f(x) note that x^2- 1= (x-1)(x+ 1) so that (x^2-1)/(x-1)= x+1 for all x &amp;lt;b&amp;gt;except&amp;lt;/b&amp;gt; x= 1. Since the limit as x goes to 1 does not depend on the value at x= 1, the limit of (x^2+ 1)/(x-1) is the same as the limit of x+ 1 which, it is easy to see, is 1+ 1= 2.