A limit asks what a function is heading toward, not where it lands. That one idea defines the derivative in Unit 2 and the integral in Unit 6, and it is how calculus handles change "at an instant".
An average rate of change needs an interval; at a single instant it would be $\frac{0}{0}$. A limit fixes that: $\lim_{x \to c} f(x) = L$ means $f(x)$ can be made as close to $L$ as you like by taking $x$ close enough to $c$, **but not equal to** $c$. What happens exactly at $c$ doesn't matter.
Read a limit from a graph or table by checking where the values head from **both** sides. One-sided limits $\lim_{x \to c^-}$ and $\lim_{x \to c^+}$ must agree for the limit to exist. A limit fails to exist if the function is unbounded, oscillates, or has different left and right limits.
Try direct substitution first. If you get a number, that's the limit (for continuous functions). If you get $\frac{0}{0}$, the limit may still exist: factor and cancel, multiply by a conjugate, or use a trig identity, then substitute again.
If $g(x) \le f(x) \le h(x)$ near $c$ and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$. It handles limits algebra can't, like $x^2 \sin\left(\frac{1}{x}\right)$ at $0$.
$f$ is continuous at $c$ when $f(c)$ exists, $\lim_{x \to c} f(x)$ exists, and they're equal. Discontinuities are removable (a hole), jump, or infinite (a vertical asymptote). A removable one can be fixed by redefining one value; a piecewise function is continuous at a boundary when both pieces and the function agree there.
If $f$ grows without bound as $x \to c$, write $\lim_{x \to c} f(x) = \infty$ and there's a vertical asymptote at $x = c$. If $f(x) \to L$ as $x \to \pm\infty$, there's a horizontal asymptote $y = L$. For rational functions, compare degrees; for growth, exponentials beat polynomials, which beat logarithms.
If $f$ is continuous on $[a, b]$ and $d$ is between $f(a)$ and $f(b)$, then $f(c) = d$ for some $c$ in $[a, b]$. It guarantees that a value is hit (often a root), but not where, and it says nothing without continuity.