Calculus - Limits and Convergence

Intuition of a Limit: A limit describes the value that a function $f(x)$ approaches as the input $x$ gets closer and closer to a specific point $a$. Visually, this is represented by following the curve of a graph from both the left and right sides toward $x = a$ to see if the $y$-values converge on a single height $L$, even if there is a 'hole' or open circle at that exact point.

One-Sided Limits: A limit can be approached from the left ($x \to a^-$) or the right ($x \to a^+$). Visually, if you trace the graph from the left and it ends at height $L_1$, but tracing from the right ends at height $L_2$, and $L_1 \neq L_2$, the graph exhibits a 'jump discontinuity' and the general limit does not exist.

Existence of a Limit: For a limit $\lim_{x \to a} f(x)$ to exist, the left-hand limit and the right-hand limit must be equal. On a graph, this means the two 'branches' of the function must meet at the same vertical level at $x = a$.

Continuity: A function is continuous at a point $x = c$ if three conditions are met: $f(c)$ is defined, the limit as $x \to c$ exists, and the limit equals the function value. Visually, a continuous function can be drawn without lifting your pencil, showing no breaks, jumps, or vertical asymptotes.

Limits at Infinity and Asymptotes: These describe the 'end behavior' of a function. If $\lim_{x \to \infty} f(x) = L$, the graph approaches a horizontal line $y = L$, known as a horizontal asymptote. If the function value grows without bound ($x \to \pm \infty$) as $x$ approaches a finite value $c$, the graph features a vertical dashed line called a vertical asymptote.

Indeterminate Forms: When substituting a value results in $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the limit cannot be determined immediately. This usually indicates a 'hole' in the graph. Algebraically, this is resolved by factoring, rationalizing, or simplifying to reveal the limit's true value.

Convergence of Sequences and Series: A sequence converges if its terms approach a finite limit $L$ as $n \to \infty$. For a geometric series, convergence only occurs if the common ratio $|r| < 1$. Visually, the sum of a convergent series approaches a horizontal 'ceiling' or limit, while a divergent series grows infinitely or oscillates.