Consider the function

$\backslash [$

f$($x$)=\backslash$frac$\{$x$^\{2\}-1\}\{$x$-1\}$

$\backslash ]$

$1.$ Intuitive Understanding: Before calculating anything formally, discuss what you think happens to $\backslash ($ f$($x$)\backslash )$ as $\backslash ($ x $\backslash )$ approaches $1.$ Why might there be an apparent issue at $\backslash ($ x$=1\backslash ),$ and how can you interpret the behavior of the function near that point?

$2.$ Applying Limit Laws: Use algebraic manipulation $($factoring$,$ simplifying$)$ and the limit laws to evaluate $\backslash (\backslash$lim $\_\{$x $\backslash$rightarrow $1\}$ f$($x$)\backslash ).$ Show each step clearly to illustrate which limit laws apply.

$3.$ Connection to Real$-$World Context: How would you explain the concept of a limit and these laws to someone seeing this for the first time, perhaps in a real$-$world situation involving rates, trends, or physical quantities approaching a specific value?