Limits & Continuity

Limits form the bedrock of calculus.
The limit of a function f(x) as x approaches a value 'a' is denoted lim(x→a) f(x) = L, meaning f(x) can be made arbitrarily close to L by choosing x sufficiently close to a (but not equal to a). This is the $\epsilon$-$\delta$ definition: for every $\epsilon$ > 0, there exists $\delta$ > 0 such that 0 < |x - a| < $\delta$ implies |f(x) - L| < $\epsilon$.

Left-Hand Limit (LHL): lim(x→a-) f(x) = lim(h→0+) f(a - h), where h > 0. Right-Hand Limit (RHL): lim(x→a+) f(x) = lim(h→0+) f(a + h), where h > 0. Existence of Limit: lim(x→a) f(x) exists if and only if LHL = RHL.

Indeterminate Forms: $\frac{0}{0}$, infinity/infinity, 0 * infinity, infinity - infinity, $0^{0}$, 1^infinity, $infinity^{0}$. These require special techniques such as L'Hopital's Rule, algebraic manipulation, or standard limits.

Standard Limits (must memorize):

• lim(x→0) sin(x)/x = 1 (x in radians)
• lim(x→0) tan(x)/x = 1
• lim(x→0) (e^x - 1)/x = 1
• lim(x→0) (a^x - 1)/x = ln(a), where a > 0, a != 1
• lim(x→0) ln(1 + x)/x = 1
• lim(x→0) (1 + x)^(1/x) = e
• lim(x→infinity) (1 + 1/x)^x = e
• lim(x→a) (x^n - a^n)/(x - a) = n * a^(n-1), where n is any real number

L'Hopital's Rule: If lim(x→a) f(x)/g(x) yields $\frac{0}{0}$ or infinity/infinity, then lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x), provided f and g are differentiable near a and g'(x) != 0 near a.

Sandwich (Squeeze) Theorem: If g(x) <= f(x) <= h(x) near x = a and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L.

Continuity: A function f is continuous at x = a if: (i) f(a) is defined, (ii) lim(x→a) f(x) exists, (iii) lim(x→a) f(x) = f(a). If any condition fails, f is discontinuous at a.

Types of Discontinuity:

• Removable: Limit exists but f(a) is undefined or f(a) != limit. Can be "repaired" by redefining f(a).
• Jump (Non-removable of first kind): LHL and RHL both exist but are unequal. Jump = |RHL - LHL|.
• Infinite (Non-removable of second kind): At least one of LHL or RHL is infinite.
• Oscillatory: Function oscillates (e.g., sin(1/x) near x = 0).

Intermediate Value Theorem (IVT): If f is continuous on [a, b] and k is between f(a) and f(b), then there exists c in (a, b) such that f(c) = k. This is used to prove existence of roots.

Properties of Continuous Functions: Sum, difference, product of continuous functions are continuous. Quotient is continuous where denominator is non-zero. Composition of continuous functions is continuous.

The key problem-solving concept is recognizing the indeterminate form and selecting the optimal technique: algebraic simplification for polynomial/rational forms, standard limits for trigonometric/exponential expressions, and L'Hopital's Rule as a fallback when direct methods are cumbersome.