Calculus - Limits, Continuity and Differentiation from First Principles

Limit Definition: A limit $\lim_{x \to a} f(x) = L$ describes the value $f(x)$ approaches as $x$ gets arbitrarily close to $a$ from both sides. On a graph, imagine a point moving along the curve from the left and another from the right; if they both target the same $y$-value height $L$, the limit exists. Note that the function does not need to be defined at $x=a$ for the limit to exist.

One-Sided Limits: Limits can be evaluated from the left (denoted $x \to a^-$) or the right (denoted $x \to a^+$). Visually, these represent the paths taken toward a specific x-coordinate from different directions. For a two-sided limit to exist, these two paths must converge at the exact same vertical point.

Continuity: A function is continuous at $x=a$ if three conditions are satisfied: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$. On a graph, this appears as an unbroken line or curve. If you have to lift your pencil to draw the function, it is discontinuous at that point (due to a hole, jump, or asymptote).

Removable vs. Non-Removable Discontinuities: A removable discontinuity occurs when a limit exists but is not equal to the function's value, visually appearing as a single 'hole' in the line. A non-removable discontinuity, like a vertical asymptote or a jump, occurs where the left-hand and right-hand limits do not match or where the function goes to infinity.

Gradient of a Secant Line: The average rate of change between two points $(x, f(x))$ and $(x+h, f(x+h))$ is represented by the slope of the secant line. Geometrically, this is a straight line cutting through the curve at two distinct points. The formula for this gradient is $\frac{f(x+h) - f(x)}{h}$.

Differentiation from First Principles: The derivative $f'(x)$ is the instantaneous rate of change or the gradient of the tangent line at a specific point. It is found by taking the limit of the secant gradient as the horizontal distance $h$ between the two points shrinks to zero. Graphically, as $h \to 0$, the secant line rotates until it just touches the curve at a single point, becoming the tangent.

Tangent Line Properties: The tangent line at a point $(a, f(a))$ has a gradient equal to $f'(a)$. It represents the 'slope' of the curve at that exact moment. If $f'(a) > 0$, the graph is rising at that point; if $f'(a) < 0$, it is falling; and if $f'(a) = 0$, the graph is momentarily flat (a horizontal tangent).