what does the symbol delta(the triangle symbol)x mean?
Delta, as Galactus said, represents change in x. In your case, it represents the change of x as it approaches zero. This is the derivative at the point x = 2. Also, you made a small typo:
$\displaystyle \lim_{\Delta x \to 0} \frac{f(2+\Delta x ) \color{red}- f(2)}{\Delta x}$
Let $\displaystyle h = \Delta x$
$\displaystyle \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}$
Our objective here is to get rid of the h in the denominator so we can find the limit. Now, let's evaluate f(2+h) and f(2)
$\displaystyle f(2+h) = (2+h)^2 - 2(2+h) - 1$
$\displaystyle f(2) = (2)^2 - 2(2) - 1$
Now simply replace and simplify. You'll find that you can cancel the h in the denominator after all.
$\displaystyle \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}$
$\displaystyle \lim_{h \to 0} \frac{(2+h)^2 - 2(2+h) - 1 - (-1)}{h}$
$\displaystyle \lim_{h \to 0} \frac{4 + 4h + h^2 - 4 -2h - 1 +1}{h}$
$\displaystyle \lim_{h \to 0} \frac{4h + h^2 -2h}{h}$
$\displaystyle \lim_{h \to 0} \frac{h^2 +2h}{h}$
$\displaystyle \lim_{h \to 0} \frac{h(h+2)}{h}$
$\displaystyle \lim_{h \to 0} (h+2) = 2$