Calculus Essentials

The limit of a function f(x) as x approaches a value c is the value that f(x) gets closer and closer to as x gets closer to c. We write this as lim(x→c) f(x) = L, meaning f(x) can be made arbitrarily close to L by choosing x sufficiently close to c.

For every ε > 0 there exists a δ > 0 such that if 0 < |x − c| < δ, then |f(x) − L| < ε. This formal definition precisely captures the idea that f(x) can be made as close to L as desired by restricting x to be close enough to c.

A function f is continuous at c if three conditions hold: (1) f(c) is defined, (2) lim(x→c) f(x) exists, and (3) lim(x→c) f(x) = f(c). Intuitively, you can draw the graph through c without lifting your pen.

If g(x) ≤ f(x) ≤ h(x) for all x near c (except possibly at c), and lim(x→c) g(x) = lim(x→c) h(x) = L, then lim(x→c) f(x) = L. This is useful for evaluating limits of functions trapped between two others that share the same limit.

A one-sided limit considers the behavior of f(x) as x approaches c from only one direction. lim(x→c⁺) f(x) is the right-hand limit and lim(x→c⁻) f(x) is the left-hand limit. The two-sided limit exists only if both one-sided limits exist and are equal.

If f is continuous on the closed interval [a, b] and N is any number between f(a) and f(b), then there exists at least one value c in (a, b) such that f(c) = N. This theorem guarantees that continuous functions take on every value between their endpoint values.

The limit equals 1. This is a fundamental trigonometric limit often proved using the Squeeze Theorem with geometric arguments involving the unit circle. It is essential for deriving the derivative of sin(x).

Writing lim(x→c) f(x) = ∞ means that f(x) grows without bound as x approaches c. The limit does not actually exist as a finite number; the notation indicates that f(x) increases past every positive value as x nears c.

A removable discontinuity at x = c occurs when lim(x→c) f(x) exists but either f(c) is undefined or f(c) ≠ lim(x→c) f(x). It can be "removed" by redefining f(c) to equal the limit, making the function continuous at c.

Divide every term in the numerator and denominator by the highest power of x in the denominator. Terms with x in the denominator approach 0, leaving only the leading coefficients. If the degrees are equal, the limit is the ratio of leading coefficients.

The derivative of f at x is defined as f′(x) = lim(h→0) [f(x+h) − f(x)] / h, provided this limit exists. This measures the instantaneous rate of change of f at the point x.

f′(a) = lim(x→a) [f(x) − f(a)] / (x − a). This form is equivalent to the standard definition and is sometimes easier to apply when computing the derivative at a particular point.