Limits and Continuity

A limit describes where a function is heading as its input approaches some value — regardless of what happens exactly at that value.

We write

to mean: as $x$ gets arbitrarily close to $a$ (from either side), $f(x)$ gets arbitrarily close to $L$. Crucially, $x$ never has to reach $a$ — the limit is about the approach, not the destination.

Many limits are found by direct substitution when the function behaves nicely: $\lim_{x\to 4}(3x-5) = 3(4)-5 = 7$. The interesting cases are where substitution gives $\tfrac{0}{0}$ and we must simplify first.

Worked example. Evaluate $\lim_{x\to 2}\dfrac{x^2-4}{x-2}$.

Substituting $x=2$ gives $\tfrac{0}{0}$ — undefined, but not hopeless. Factor the numerator:

Now substitute: $\lim_{x\to 2}(x+2) = 4$. The original function has a hole at $x=2$, yet it still heads towards 4.