The Foundation of Calculus: Limits and Continuity

In calculus, a limit describes the behavior of a function as the input $x$ gets closer and closer to a specific value $a$. Crucially, a limit is about the approach, not the arrival. We write this as $\lim_{x \to a} f(x) = L$, which means as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$. If the function approaches different values from the left ($x \to a^-$) and the right ($x \to a^+$), the limit does not exist (DNE). For a limit to exist, the left-hand and right-hand limits must be equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.

Sometimes, direct substitution results in $\frac{0}{0}$, known as an indeterminate form. This doesn't mean the limit doesn't exist; it means the 'hole' in the function is hiding the limit. To find it, we use algebraic manipulation. The most common technique is factoring, where we cancel out the term causing the zero in the denominator. Other techniques include rationalizing (using conjugates for square roots) and simplifying complex fractions. These methods reveal the 'intended' value of the function at that point.

A function is continuous at a point $x = a$ if there is no 'break' in the graph. Formally, three conditions must be met: 1. $f(a)$ is defined (the point exists). 2. $\lim_{x \to a} f(x)$ exists (the left and right sides meet). 3. $\lim_{x \to a} f(x) = f(a)$ (the limit matches the point).

If any of these fail, the function is discontinuous. We classify these as Removable (a hole), Jump (the graph leaps to a new value, common in piecewise functions), or Infinite (the graph heads toward an asymptote).